What does a .app file actually do?

"apps" are "Package Bundles" is Apple jargon, and are treated specially by the OS. As mentioned above, these are actually Unix directories, in a special format. The actual Unix Executable File is in a subdirectory named MacOS, which you can see using the "Show Package Contents" menu in Finder.

The "app" is analogous to the directory created when you install a Windows program. You can indeed navigate into the app, and run the Unix Executable File.

For more complete details see the Apple Bundle Programming Guide It is in the archive section but Apple's new documentation does not give the overview

This doesn't solve your problem, but without knowing more about the app it won't be solved by manipulating the package. You will need to contact the developer.

You could also look at the Console app, which will let you examine any error or warning messages.