I don't see how Qt does the low level graphics work, in order to create its own custom GUI look/feel for each platform. Does it utilize each of the platforms APIs or something? I ask because I am really wondering how I could go about creating my own framework in order to make a custom GUI application with a unique graphical look.
The good people at Trolltech (now Nokia) write all the platform specific code for you. Only the interfaces to their library remains the same across platforms. You do have to make sure that you only use their classes and don't make any OS-specific calls. If you want, you can modify the graphics and the handlers to extend the Qt library and create your own look and feel.
"How does QT only use C++ to make custom GUIs for some many platforms?"
"Does it utilize each of the platforms APIs ... ?"
You answered your own question. It makes a common library for developers to write their applications with. The application developers write their application only once, not caring about platform specifics. Then, the library author puts out different versions of the library for each platform, which handles all the specific UI calls. This is called encapsulation:
http://en.wikipedia.org/wiki/Information_hiding#Encapsulation
If you were to write your own such library, you would need to figure out what is common between all of the platforms you target, or figure out what high-level concepts your application would need, and create that abstraction. Then, implement that abstraction for each of the platforms you wanted to support.
Edit:
Also see Juliano's comment. This seems like it might be closer to what you want to do, rather than the question you asked. I'd not re-do the work that Trolltech or the WxWidgets team or the Gtk guys or the Mono people did, if I could possibly avoid it.
Your Qt application doesn't have to have the plain standard platform graphical look, but there are ways to design unique apps with Qt.
For instance, you can style your application with style sheets: http://doc.trolltech.com/4.6/stylesheet.html, or even with a custom QStyle (which is not very easy): http://doc.trolltech.com/4.6/qstyle.html.
Qt Quick is a new technology we're introducing in the upcoming Qt 4.7 (check out the beta if you're interested). It allows you to easily define custom UI components from simple primitives, with custom animations, state transitions and other effects. See for example http://www.youtube.com/watch?v=8G4U7QWRajg.