Watching pages allows any logged-in user to keep a list of "watched" pages and to generate a list of recent changes made to those pages (and their associated talk pages). In this way you can keep track of and react to what's happening to pages you have created or are otherwise interested in.
Watching a page means that recent changes made to it show up on your watchlist (Special:Watchlist). Its associated talk page goes onto your watchlist, and vice versa. Actions affecting watched pages (page moves, page creations and deletions, protection) also appear in the watchlist. For example, if you watch a page that does not yet have a talk page, you will see on your watchlist when someone creates that talk page.
If a page you have watched is moved to a new title, the new title will be automatically added to your watchlist. Even if the page is later moved back (and even if the page at the new title is deleted), the new title will remain in your watchlist along with the old one. If you notice mysterious nonexistent pages appearing on your list of watched pages, this is the most likely explanation.
When you view Recent Changes, entries relating to pages you are watching appear in bold. This means that it may be beneficial to mark pages as "watched" even if you do not intend viewing the Special:Watchlist page.