It might also help to recognize that the concept of "distance" in general relativity is much more subtle than the everyday version because it depends on the coordinate system. GR gives you the instructions for how to get a distance once you have chosen a coordinate system, which means GR is a "metric" theory. We are not used to this, we think you can use 2D latitude and longitude coordinates, or Cartesian coordinates in 3D, and still get that the distance between New York and London is a particular thing. But on scales of the whole universe, you can get very different distances depending on your means of coordinatizing the events. Because of the cosmological principle, however, there is one coordinate system that stands out, called "comoving frame coordinates" that move with the average of the local matter, because in those coordinates the universe seems to be the same everywhere at a given age. It is that choice of coordinates that parses between "what space is doing" and "what matter is doing within the space." Whether this is just a convenient language, or represents something deeper, requires the discovery of new laws to determine, but right now the cosmological principle is a kind of organizational principle of convenience.
Note however that the speed limit of special relativity is not subject to these problems, because that speed limit only applies to two objects passing each other at essentially the same place and time. The global coordinates don't matter in such a local interaction, just like you don't need latitude and longitude coordinates to tell how fast you are passing someone on the highway.