MAKERphone is a class (type) that encapsulates all the unique functionality of the MAKERphone. This functionality is provided by member functions and objects. In this case, the member functions you mentioned do the following:
begin() - Initializes the object (a MAKERphone) and tells it whether to display the moving splash screen during initialization or not.
update() - Called periodically, this syncs the hardware with the software library. This is rather time consuming and should not be called unnecessarily.
loader() - restarts firmware if it is still available in the other memory partition.
while the object members (in your case, display and buttons) provide their own member functions to affect sub-parts of the MAKERphone:
display.print() - for example, write text to display (overloaded to support string type and assorted numeric type arguments)
buttons.released() - for example, returns whether a specific button has been released.
If you read MAKERphone.h, you’ll see lots more members of the MAKERphone class, and for those members that are objects (of a particular other class), those objects will have their own member functions and object members. And so on and so on.
I agree this is a TERRIBLE way to try to understand the MAKERphone library, so hopefully CircuitMess will soon publish a library reference manual similar to that for other Arduino libraries.