What is this hook supposed to do? I need to modify the query of one of my views. I've found hook_views_query_alter(), but can't for the life of me figure out what I'm supposed to do with it. According to the API, 


  This hook should be placed in MODULENAME.views.inc and it will be
  auto-loaded. MODULENAME.views.inc must be in the directory specified
  by the 'path' key returned by MODULENAME_views_api(), or the same
  directory as the .module file, if 'path' is unspecified.


OK, so I know to put the function with the hook in a views.inc file. But what do I need to put in the actual .module file? I can't seem to find any examples of complete module code that uses this hook to point me in the right direction. Does my module code contain the MODULENAME_views_api() and nothing else?
If your module doesn't need to implement any other hooks, then yep, hook_views_api() is all you need in your .module file. 

As long as you provide the right path in hook_views_api(), Views will know where to look and will dynamically include the files when they're needed.

Usually you need to put files[] = mymodule.FILENAME.inc in your module's .info file for the dynamic code registry to pick it up, but from the looks of it that isn't the case for Views.

If you're looking for some code examples for this, take a look at the Administration Views module; it's .module file contains only hook_views_api() as well (and it works), so you can be confident you're doing the right thing.