How is the compiler built? The same way standard non-web compilers are built. Really, the web interface is just a interface to the compiler -- there's nothing fundamentally different about compiling a program submitted via the web vs submitted from the filesystem.
What would be fundamentally different would be if the service allowed you to compile and run the program on the server. Running untrusted code on the server poses some challenges (how do we do it securely?). If that's what you wanted to know about, Realz Slaw's answer should provide some more details about that aspect. But in your question you only asked about compiling, and there's nothing different about compiling.