For any still confused, there is also a 'raisin bread' model (which I prefer because it's 3D, unlike the surface of a balloon). The raisins are galaxies, and the bread is the space between them. As it bakes, the raisins get further apart, even though the raisins aren't moving through the dough.
(Although galaxies also have movement through space, this isn't what is causing them to get further apart.)