More space. It's pretty nuts. Imagine you were stretching a rubber band. What's in-between two points on that rubber that was stretched? Just rubber, albeit thinner and less dense.

In our universe, however, the space that's created is the same as the space that was just there. It'd be like if the rubber band stretched, but didn't lose any density while it did so.

Insane, I know.

The string won't break apart, because the forces that are holding it together are so much stronger than the "force" that is pulling our universe apart.

Unfortunately, it doesn't answer your more fundamental question of "if I make 2 marks in space, and then those expand apart, what is in the new space, and how do I know they really are farther apart?"

I have never heard a good description of this on human scales, only at stellar and galactic scales.