You'll find yourself becoming more responsible. One of the main reasons you should live off-campus, at least for a couple years, is that it offers important real-life experience you can build on as you move into post-grad life.
Between actually having to pay your own rent (which most students don't have to worry about in the dorms), and managing all the other expenses that go along with renting your own place, living off-campus is a crash course in responsibility. Things like time management, budgeting and learning to cook will all start to become normal parts of your day-to-day life. Before you know it, you'll be a full-fledged adult.
You'll also learn how to develop a relationship with your landlord, and you'll build a rental history, which will only help you when apartment hunting in the future.