Here in Ithaca NY, Collegetown used to be a sleepy neighborhood of large divided houses rented to Cornell students.
About 20 years ago they began densifying like crazy. 2-3 story houses for 8 people became 8-floor apartments and studios. And you know what? Rent is higher than ever! Everywhere!
Once you build up you can charge higher rents, which means the landlord next door can charge slightly higher rents, and so on.
Building housing almost never lowers rents for the non-rich, unless there is a massive recession (Florida), or an equivalent rise in wages. People build dense housing because they expect people to keep coming...