In a finite universe, what happens when light reaches the boundary? If a universe is finite and is not expanding at a speed equal to or greater than c, what happens when light or another form of electromagnetic radiation traveling reaches the boundary?
One important concept that has not been mentioned yet is the "cosmological principle."  This is the key simplifying idea for our cosmological models, it says that the universe is the same everywhere on the largest scales at a given age, so alien astronomers 50 billion light years from us that also conclude the universe is 13.8 billion years old will be observing pretty much all the same things we are.  This principle is not proven by data, but the data is consistent with it, most notably the Hubble law and the homogeneity of the cosmic microwave background.  It allows us to understand the past of our own part of the universe by seeing the past of distant parts (a constraint we are stuck with given the speed of light).

So that's why a universe model that is finite in size must curve back on itself-- nothing else would satisfy the cosmological principle.  This doesn't mean the principle is true, it means we are not going to part with it unless we have to.