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Money, traditionally.

Unless the author is relatively well known, the book tour often barely pays for itself. In some cases the author gets to travel "on the company dime" and basically tour the country/world and have the expenses paid by some book sales.

The author may also get a much larger cut of any books sold during the tour, even without "signing fees" or similar.

They are hoping to get the big break. That can be people who see them locally buying a copy (most people will buy a book when the author is there, but may not even look at the book otherwise, so you can count on a few sales if you appear in a bookstore), but ideally is moves onto they actually read the book and like it enough to tell all their friends. You can get a big bonus if a someone well known happens to be in the crowd and tells all their followers.