This is incredibly cool. The signal is hard to detect because it's quite faint and there are tons of other signals superimposed on it: the galactic synchrotron, other astronomical sources, and terrestrial radio (FM, for example). And the neutral hydrogen involved can both emit and absorb 21cm light. Getting it right is hard, but various groups are working on it.

Once we can not just detect the signal but measure its spatial and frequency variation, we'll get a very nice map of the earlier universe. In particular, the 21cm line is very narrow, so its power spectrum as observed from Earth gives the strength of the signal as a function of redshift, which is strongly related to distance. If we can get some decent spatial resolution, this will give us a 3D map of the density of neutral hydrogen on the early universe.

This will give great data for cosmological calculations. Most useful cosmological observations right now are either 2-dimensional (e.g. the cosmic microwave background, which we can resolve angularly with great detail but represents a very narrow slice of time) or barely 3-dimensional (supernova observations, for example, which let us map a much smaller range of times than we can potentially map with 21cm observation).

(Part of my thesis was on an approach to detecting the 21cm signal and its spatial variations.)

Followup: A friend who works on HERA, which is also looking for the 21cm signal, forwarded this discussion: https://twitter.com/UCBProf/status/969071237405097985