You wouldn’t be renting. College students live in college residences (“dorms”). You’re right they’re very expensive in NYC and for NYU - but the comprehensive fee includes tuition&fees (cost of classes, using library&labs, office hours, support centers, health centers, athletic facilities…) + room and board (a room in a college residence, shared with another student the same age as you and generally same gender unless you request differently, sometimes sharing academic or extracurricular interests if you join “living learning communities” + access to 3 meals, with unlimited food, each day.)
Williams includes these costs, as well as the cost of books, transportation, and clothes, when they determine how much it’d cost your family then how much they need to give you for you to attend.
NYU doesn’t “meet need” so you never know how much they’ll decide to give you -often, it’s not enough to pay for the university costs, also known as a soft rejection.