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What role does an API serve in MVC?

I'm still trying to understand MVC to it's fullest.
I know that in a normal setup, the user requests a webpage. That request gets read by the controller and if for example the user needs to see 5 blogposts, it asks the Model to retrieve 5 blogposts out of the database and give those back to the controller. The controller then sends those 5 blogposts to the View, where they get inserted into a normal webpage. The View then returns the data to the user and they see 5 blogposts on the website. (Please correct me if I am wrong. I find it really hard to grasp, for some reason.)
Now I am using React in a project and I would like to do API calls to my PHP to retrieve data.
Does this mean that if I request "website.com/api/blogposts/5", the Controller in PHP reads that it's an api call, asks the Model for 5 blogposts and sends this data back as a response. This would make React the View, right?
Sorry if this is supposed to be basic information. I just can't wrap my head around it.

It seems like you might be conflating a few concepts so let's take a second to separate them.
Let's talk about web API's first, based on the context you've given you're trying to build a react app that communicates with a PHP API. MVC doesn't necessarily apply here. It exists as a separate application so trying to think of them together in some kind of MVC architecture is going to cause confusion. This API could feature an architecture like REST and would be built to serve data to your client-sided application.
React, my knowledge of the platform isn't extensive however in theory you could map out some type of MVC style architecture in how you structure your project, but React isn't an MVC framework. It's very component based and designed to create reusable components for single page applications. You can read more about this here.
MVC, I find it easiest to explain using asp.net mvc as an example but I'll try to explain a way that doesn't depend on that. MVC is a principle seen most in the typical server-sided web page loading platform. The view being the web page, the controller being a route on a server, and the model defining the data seen on the page which was returned by the server. Most of your MVC guides will explain this pretty thoroughly.
To summarize, if you're trying to learn React for your front-end applications and design an API in PHP to use with it, you'll want to learn React how it was designed to be used. For the API, take a look at good REST practices and just general PHP API designs. If you want to learn MVC, checkout an MVC framework that is built on top of that principle. The concept has some age to it now, so some of the newer technologies like React don't really use the same principles.

Does this mean that if I request "website.com/api/blogposts/5", the Controller in PHP reads that it's an api call, asks the Model for 5 blogposts and sends this data back as a response. This would make React the View, right?
This could be an approach. However, since i've done this before I strongly recomend not to put the API logic inside the controllers. If your application tends to scale, your controller will become huge, and that is something you want to avoid.
What you should probably do is to create a restful implementation that handles the requests and validations, and from there you can call your controllers to retrieve the processed data and then answer the request. With this approach you will get more maintanable and consistent code.
And of course, you will keep your MVC pattern implementation clean and pure.

API, back-end and front-end as all three separate components

I tried to find something on the internet but could not find anything similar. So I'm asking it here:
SITUATION: I have a big API which does some heavy calculations and has a lot of functionality. There are some clients using this API and has implemented it in their software. Now I want to write some front-end for that API so some users could manage their workflow more easily.
CONSIDERED SOLUTION: I am considering of making a separate back-end application which would use an API and serve for the front end (look at the picture attached). The backend would do authorization / caching / data-adapting operations.
QUESTION: But I have never ever crossed such app design with three layers API-BE-FE. So is it worth making things this way? Are there any significant drawbacks? Is it safe to put some oauth authorisation in the back-end side, not api itself? Like what are your thoughts about it?

I agree with your design. You have a specific API which is meant to serve specific endpoints. This way you are separating your concerns, as you can add to your BE things that aren't related to the API itself, but are related to the FE.
Also, many APIs are using credentials and keys so you can implement a similar functionality.

Your considered solution on architecture looks good.
The most biggest advantage to implement a back-end between front-end and API is, it can provide good separation of concerns. It usually happens around me that front-end engineers ask API engineers every time when they need new endpoints. It looks just cooperation, but sometimes goes too much. This kind of conversation has potential to result in making too many endpoints in API which shouldn't have had. I am not really sure what the architecture policy of API team in your company is, but just to allow API to be growing big for front-end is not good. The more functionalities the API has now, the worse it will easily be.
In your plan, you are trying to implement back-end to access API for front-end. It was similar to the architecture of BFF (Back-end For Front-end) described by Sam Newman (http://samnewman.io/patterns/architectural/bff/). With this concept, you can implement a back-end as a kind of a gateway which handles front-end specific requests to API. Back-end can even buffer the potential influence to API caused by change in front-end if needed. Everything can be kept well separated.
In BFF, I don't think that back-end plays a role to provide application-related functionalities such as authorization, caching, and data-adapting operations, but this depends on you. You can implement new APIs to handle those functionalities and have back-end just be a gateway which ties them up. It would also work just to put those things into back-end as long as it is not too fat.
Drawback?
The possible drawback, I suppose, is maintainability of scaling. This totally depends on the infrastructure team or members you work with, but on production, API and backend will run on each different server or stack, so you might need to take care of scaling consistency among them under the large amount of traffic to your application. However, this independency could also be advantageous in monitoring hardware resources. You'd better to find a sweet spot.

What helps store the inputted data on a backend page in HTML?

So I am new to web development and all I've learnt so far is how to write HTML and CSS to make web pages and forms.
I'm specifically looking for a language that will help me store the data that is input into the form onto a database for easy access later.
I think PHP does that, but I was looking for anything JAVA related, and somewhere I could learn how to do it.
So far,
I've looked into JavaScript tutorial on W3Schools, but it seems like JS only helps make the front end more dynamic, but doesn't help store the input data anywhere.
I've also looked into the AngularJS tutorial on CodeSchool and it looks like Angular helps present the forms in a better manner and again, not in storing data anywhere.
Once again,
1) What do you use to collect the data input into a HTML form and store it somewhere?
2) I don't need help making the actual form itself.

Me personally, i use PHP, it allows you to take the data that was inputted and store it in a database. You can also use JavaScript/JQuery with PHP via Ajax that will dynamically fetch and store data.

Any backend lang can do it (python, php, java, ruby, js with node), but each one has it own ways to do it. Even JS at client side can do it, using LocalStorage objects, but it not solve all problems at data storage yet.

Angular is just the client-side. You will need a server, which will handle the requests and store them into a database. I would work with Spring Boot (JAVA) on the server side, and Angular 2 on the client side. Use REST for the communication bewtween them. Its really not hard.

Node.js is a fast Javascript runtime combined with a low-level API similar to the standard libraries of many programming languages (file system access, buffers, streams, i/o, etc.).
Angular is a model-view-controller framework for client-side JS development. It can be used with Node.js as a backend, or anything else. Its main feature is 2-way data-binding, and addresses most of the concerns of a single-page web app within the framework.
React is often compared to Angular, because it is a front-end library, but it is not a framework. It is simply the view layer, with a large ecosystem of open-source projects supporting it. The big conceptual difference is a uni-directional data flow, rather than 2-way data binding. You need a lot more besides React to make a full application, but React handles it's use case exceptionally well. React is amazing on the client side, but it’s ability to be rendered on the server side makes it truly special. This is because React uses a virtual DOM instead of the real one, and allows us to render our components to markup. Node.js makes a great backend for React as well, but again, it can work with any backend.
The MEAN stack is a popular web development stack made up of MongoDB, Express, AngularJS, and Node.js. MEAN has gained popularity because it allows developers to program in JavaScript on both the client and the server. The MEAN stack enables a perfect harmony of JavaScript Object Notation (JSON) development: MongoDB stores data in a JSON-like format, Express and Node.js facilitate easy JSON query creation, and AngularJS allows the client to seamlessly send and receive JSON documents.
MEAN is generally used to create browser-based web applications because AngularJS (client-side) and Express (server-side) are both frameworks for web apps. Another compelling use case for MEAN is the development of RESTful API servers. Creating RESTful API servers has become an increasingly important and common development task, as applications increasingly need to gracefully support a variety of end-user devices, such as mobile phones and tablets.
This was the overview of all the new booming technologies.. Based on this you can decide what you need and what you want to learn.. Thanks, Hope this overview helps you to decide.

My favorite for Rest Api is Flask(python micro framework) it is build for create Rest Api. and for php falcon has it own micro frame work. if you use nodejs it is easy to communicate between backend and frontend and good with not sql dB like mongo db.

The right answer should be the database or localStorage/sessionStorage.
The decision in between which answer chose, depends if you want to share the data between Browsers/Computers or if you just want to temporarily store the data for the user so he won't need to fill the form once again.
For temporarily storage chose localStorage/sessionStorage (javascript).
For other cases chose to store the data in a Database.

Client-side or server-side framework?

My project would be a kind of craiglist, a site where users could post anouncements (evereday-life objects, cars, flat etc.). So, authentication, profile page, content creation, display the for-sale objects etc.
I have developed a very large part of the backend: I have a RESTful API in three-tier architecture developed in java. It makes the link with the db, to provide me with different urls and send me the relevant JSON.
URLs example:
http://api.mywebsite.fr/user?userid=1 sends me back:
{"user": {"username": "jdoe1234", "email", "jdoe1234#gmail.com"}}
I have urls for all actions performed on the entire site (anouncement creation, last data updates ... everything, and I've carefully declared them POST, GET, UPDATE, DELETE, etc.). There is also oAuth to protect the API from queries that are not allowed for the token.
That's all for the "server" aspect, I think that there is no problem with that.
But if all the actions are managed by the webservice, I do not see the interest that could bring me a big server-side framework like Symfony/cakePHP, Zend, etc., to make HTTP requests on my different entry points, retrieve JSON and populate the HTML.
So I looked at client framework, like Angular, Ember and so on. At first, it seemed very suitable for my case: possibility of http requests, manage what to do in case of success or error, directly exploit the resulting JSON to populate the view etc.
I didn't even manage to make my choice between angularjs and Ember, both being very similar, but with the release of Angular v2, I fear the maintainability of v1 (if I choose Angular, it will be v1 , because the majority of tutorials and questions relate to Angular 1.X).
I don't know if I'm doing the right thing by choosing client-side framework, I am afraid that they 'brident' (not sure of that word, sorry) me. Plus, it's fully instantiated in the browser, so the user can change absolutely all code and data I provide. That seems weird to me.
I want to be absolutely sure of the technology that I use in case I make this application available to the public for example. I want to do things properly, in order to avoid maintainability or security problems.
Summary: With the things I already have (webservice / api), is it a good idea to use a client framework like Angular or should I stay on big server-side framework like Symfony/Zend etc? Knowing that I position myself in the context in which this platform would be massively used (craiglist comparable traffic).

I'd say - depends whether you want to be more frontend guy or backend guy in future. If you want to be full stack developer then it doesn't apply.
In my opinion, both Symfony/Zend or other big server-side frameworks aren't so exciting as dynamic frontend JavaScript frameworks like Ember/Angular/React.
Also, if you have already RESTful API and OAuth authentication implemented in backend part I'd go with Ember. Why? Ember Data is great tool for talking to backend API. It's mature, it lazily loads records when they're needed and it's very customizable.
it's fully instantiated in the browser,so the user can change
absolutely all code and data I provide...
Ember has built in security like sanitizing data which is rendered in it's templating language - HTMLBars. Also, there's CORS and content security policy (CSP) standard which is implemented in Ember.
I want to be absolutely sure of the technology that I use in case I
make this application available to the public for example. I want to
do things properly, in order to avoid maintainability or security
problems .
In Ember you can create mature, secure, production-ready applications, but you need to comfortable with your Ember skills to some degree to build such ambitious web application, but it's part of building every application.
With the things that i already have(webservice / api), is it a good
idea to use a client framework like Angular?
Yes, it's very popular solution to use MEAN stack or go with Ember + RESTful API.
Why should I choose Ember instead of Angular (which have a larger
community/tutorials/answered questions) ?
Angular has larger community/tutorials/answered questions, but when I started some side project with Angular to learn its possible advantages over Ember, I was surprised how there was no consensus in it's community for doing one thing. So, instead of fast search how to declare and use directives (I think it was the thing that confused me) I have to do another research which way is the best. Also, there are lots of ways to setup project (where to put custom directives, different Angular objects) and you have to do another research which one to choose. I ended up using repo healthy-gulp-angular as my template, but you can see it hasn't been updated for 8 months, but I think during these 8 months Angular had a lot of changes and I'm not sure if this repo is the best choice.
In Ember you have Ember CLI tool which is built with Convention over Configuration principle. You have also Ember Data which utilizes JSON API standard - if you don't have JSON API compliant server side right now, you can write custom adapter to normalize server responses or change how backend replies. In Ember you don't have all that headache and different best solutions to do 1 basic thing depending who you ask.
What means "Single page application" ?
Single-page application is basically a page which doesn't have to reload all assets and HTML when you navigate. It's advantage over PHP - when user moves to another location he downloads only new data for that route. More info here.
Does those frameworks allow me to create real routes ? (
www.myapp/profil/userid etc )
Yes, of course. You don't even need # in your URL. With simple rewrite rule and small amount of logic for profile route and specified path profile/:userid, when user will open URL www.myapp/profile/userid he will be automatically taken to profile route, and userid would be interpreted as route parameter, so you can take this userid and find user record from the store in model hook.

Client = speed, Server = stability
JS frameworks updates once per week
Non-Js back-end once per year
Client side depends to behavior depending on browser
Back is related only on machine but not on environment
I chose FE coz I tired to debug code by writing variables values to database to actually see what is going on in controllers -_-

Single Page Application - Frontend independent of backend?

I've done some research and I've noticed that in a lot of examples Symfony2/AngularJS apps the frontend and backend are combined; for example, views use Twig.
I'd always thought that it's possible (and common practice) to create the frontend and backend separately and just join them by API. In that case if I want to change a PHP framework I will can do it without any problems and it will be enough to keep API.
So what are the best practices for doing it? It would be great if you could explain it to me and even greater if you just give me a link to good example on github or something.

We have been developing some projects using the same approach. Not only I think it doesn't have any "side effect", but the solution is very elegant too.
We usually create the backend in Node.js, and it is just an API server (not necessarily entirely REST-compliant). We then create another, separate web application for the frontend, written entirely in HTML5/JavaScript (with or without Angular.js). The API server never returns any HTML, just JSON! Not even an index structure.
There are lots of benefits:
The code is very clean and elegant. Communication between the frontend and the backend follow standardized methods. The server exposes some API's, and the client can use them freely.
It makes it easier to have different teams for the frontend and the backend, and they can work quite freely without interfering with each other. Designers, which usually have limited coding skills, appreciate this too.
The frontend is just a static HTML5 app, so it can (and we often did) easily be hosted on a CDN. This means that your servers will never have to worry about static contents at all, and their load is reduced, saving you money. Users are happier too, as CDNs are usually very fast for them.
Some hints that I can give you based on our experience:
The biggest issue is with authentication of users. It's not particularly complicated, but you may want to implement authentication using for example protocols like OAuth 2.0 for your internal use. So, the frontend app will act as a OAuth client, and obtains an auth token from the backend. You may also want to consider moving the authentication server (with OAuth) on another separate resource from the API server.
If you host the webapp on a different hostname (e.g. a CDN) you may need to deal with CORS, and maybe JSONP.
The language you write the backend in is not really important. We have done that in PHP (including Laravel), even though we got the best results with using Node.js. For Node.js, we published our boilerplate on GitHub, based on RestifyJS
I asked some questions in the past you may be interested in:
Web service and API: "chicken or egg"
Security of an API server: login with password and sessions