I haven't visited any in recent months, but I used to borrow some books from time to time. Sometimes without even knowing what exactly I'd like to borrow, I'd just go to eg. fantasy section and walk by the shelves, picking up some novels till I'd find something interesting. I don't have any e-book reader, and e-books in general are not yet as popular and widespread in my country as they are in eg. US, so libraries are the way to go if I want to read a book, but no necessarily want to buy it.
In recent years, all public libraries in my district (about 16 libraries in 24 square km area) got modernized, there's just one card with bar-code which you can use in all nearby libraries, can log online to check what books are available and where, what you got borrowed and when you should return it and so on. It also made the librarian's work a bit easier, as previously they'd have to look in alphabetically sorted shelves for someone's card (so you had to register in each library separately).