For comparison how much space does a supercluster take up? Without knowing the ratio it's hard to tell if there really is homogeneity or if they just didn't study a big enough area.
Very good question. Let's see: Superclusters and filaments are typically on the order of a few hundred million light years across, and the largest ones can be almost a billion light years long. It's not obvious how to compute the volume of these things, since they're rather aspherical, but a sphere containing one a billion LY across would have a volume of 0.5 GLY
3, which is still quite a bit smaller than what the survey looked at. The survey also examined several different portions of the sky, so I imagine it's a fairly representative sample of the universe at large.
Apparently the method for determining homogeneity is to see if the number of galaxies increases as the cube of the distance from a specific point (presumably Earth), and the data shows that indeed it does for sufficiently large distances. I assume that has to take the expansion of the universe into account, since the universe gets less dense on average over time and the farther out we look the farther back in time we see.