The foam is flexible because each tiny strand is so slender and small that bending the entire eraser is easy. In airy melamine foam, only a very limited amount of casing stays in place, and the strands that do are located where the edges of several air pockets overlapped. You can picture a squishy sea sponge as an example. For open-cell foam (typically the more flexible) imagine that those balls have burst, but that some sections of their casings still remain.

Types of closed-cell foam are usually more rigid because they retain most of their air pockets intact, like a bunch of balls all crammed together. Closed-cell foam is easier to visualize, so let's start there.