They analyze lots of music and try to make connections within and between them, using a wide variety of analytical techniques. If you're actually asking and not being sarcastic, the sibling post links to several journals that will give you a good idea what theorists tend to work on.

When I was in school it seemed like most of the theory PhDs would pick a composer to focus on, but there are other ways to go than just that. Certainly, the sorts of questions I listed (asked in much more specific ways) are part of analytical inquiry, but a lot of theory papers are descriptive rather than explanatory.

Edit: I was performance major, but took classes from some pretty great music theorists. A few examples of graduate-level classes I took: the string quartets of Bartok; the music of Stravinsky; the symphony in the 19th Century. They tended to be focused on an individual composer or repertoire. Obviously scholarship is done on a much more granular level than that.