I can only talk for myself, as I am a great fan of that: Links to the pages themselve enable you to easily post them on Social Networks, Instant Messengers and E-Mails. URL often contain unnecessary parameters, or worse: sometimes even parameters that reveal personal information about you. That can pretty much suck when you post stuff in public Chats/Forums/Blogs.
Another thing is HTTP Post: especially when you are on such user-content-driven sites like Stackexchange, you often do POST-operations. Pressing F5 after a POST operation gives you this ugly and annoying browser window: "Do you really want to reload and post..?" If you accidentally click "OK", you posted your stuff twice with even more annoying consequences.
Anyways, it makes navigation and sharing of User driven Web content much more predictable for the user. Meaning less stress and more fun.
(Sometimes I wished Google would provide links to their searches. Particularly after they introduced that instant search, rendering the browser URLs more or less useless. Pretty cumbersome when you are a fan of sharing hyperlinks.)
EDIT: last but not least: sometimes the URLs on top of the browser are not valid for an infinite amount of time. (cf. "Permalinks")
And in the age of Web apps it becomes preferrable to not see your Address bar so often. On small screen devices the user may not want to see that extra waste of space as less often as possible.