The 2D descriptions are just analogies, to make unvisualizable ideas visualizable or compare unfamiliar concepts to familiar ones.

Some theories require extra spatial dimensions from 4 on up which we can't detect or imagine ourselves, or involve some peculiar compression/stretching/bending of space in a way that's most easily visualized as happening in an extra spatial dimension. So, to try to illustrate those concepts, people simplify the world down by at least one dimension, using two to represent what we normally think of as the plain ordinary three, so that they'll have the third dimension available in which to show whatever higher-dimensional oddity they want to talk about, because they just can't use actual higher dimensions to show what they're REALLY talking about.

Another analogy using a 2D universe has nothing to do with extra dimensions but just seeks to make an unfamiliar concept more familiar anyway: the expansion of space. What's most accurate to say is that all three dimensions are stretching so the space between any two points gets longer, but most people have never seen anything expanding like that in three dimensions, so the idea of the universe as a 2D balloon surface is used instead because people are quite familiar with balloon stretching. The idea is to get people to apply what they're familiar with in 2D balloon surfaces to the less familiar 3D setting for the same concept of stretching. It doesn't mean anyone's claiming the universe is actually 2D; it just means that they're momentarily pretending, or talking AS IF it were 2D, to get audiences to take the 2D concept and then step it up to three.