Why won't Windows EXE files work on Linux?

Linux and Windows executables use different formats. Linux uses the ELF format on most architectures, while Windows uses the PE format. ELF is better suited to the way Linux manages shared libraries, and PE is better suited to the way Windows manages shared libraries, but there's no fundamental reason why Linux couldn't execute PE executables or Windows ELF executables. In fact Linux can execute PE executables, through Wine.

The difficulty is that Windows and Linux have completely different APIs: they have different kernel interfaces and sets of libraries. So to actually run a Windows application, Linux would need to emulate all the API calls that the application makes. That's a lot of work. Wine does it to some extent, but it's very hard, especially since the maker of Windows doesn't cooperate. You can compare it with, say, learning English when your native language is Chinese: the executable format is the alphabet (not that hard to master), the API is the vocabulary (takes years to get to a level where you can start reading literature).

Windows binaries have a different ABI and use a different API than Linux binaries.

Linux binaries don't need an extension because *nix uses permission bits to identify an executable instead of the extension.