First what is a RESTful API
To clear up a few things RESTful APIs (aka a service) are a means to have your applications communicate. You create a service and then many clients that can make request to it.
What is a client? These are many things like the front-end web UI, a reporting engine, the invoicing system, search and so on.
So in your example you would be building a RESTfull API service for the front-end client to make requests to. A RESTful service will send the data back usually as json. The client then process that response processing further and displaying as appropriate.
One thing to note here is yes the service is stateless but the client is not. Your client can have sessions and cookies. It will have to pass that information in the request to the service for processing using the the response to update the state.
So for example a client could make a call to routes like:
/user/create //A POST request with all the information needed to create a user /user/{:id}/profile //A GET request to find get the information needed for a //user profile with whatever the id is.
So a RESTful API is a high level architecture for the applications' communication.
Where is MVC in this?
A step down into an application is where MVC (as well as MVP, MVVM, etc) is used to create separations of concern. These patterns help so you do not get disparate piece and parts all over the place and more logic in your views then html. You would see this implemented within the front-end client because the service does not care about views just the data (view is the V in MVC).
To break down MVC:
(M)odel - is a container more or less to hold the data. It could be a user model or a larger model like profile that has a user, cart, payment, etc. It has a known structure and elements and probably some methods for data manipulations.
(V)iew - is the way to display the information back to the user (browser with HTML). This could be a php page that takes in the above model and parses it to the various section of a page. One thing that is pretty agreed on here is that this should not contain much logic. It is mostly presentation of data that was gathered.
(C)ontroller - orchestrates the application. Based on where a browser request hits it knows where to get the information to fill the model in this case a request to the RESTful service API. That model is then feed into a view and then sent back to the users browser.
Summary
The beauty of creating a RESTful service is that it can be in anything and then the client can be in anything just as long as you can make calls over http/https. The MVC is used in a client that manages how users interact with that API.
Implementations
Now as for an implementation in PHP check out some of the frameworks Laravel, codeignitor, symfony and yii all come to mind (I have used 0%) and see if any fit your needs.
Note your could home grow the service and then pick a framework for any of the clients usually I build my service in PHP or Java (PHP has slimeframework and Java JAX-RS) and then choose whatever hottness I want to use for that day in the client (Angular, React, GO, or more PHP, hopefully not Java).