ANITA is a radio telescope attached to a balloon looking for broaband impulsive radio emission in Antarctica.
The main purpose is to look for the Askaryan emission from neutrinos interacting in the ice. The Askaryan emission is just the coherent version of the same process (Cerenkov radiation) that produces the flashes of light in IceCube (basically at long wavelengths you can't resolve the charges in a cascade and see a fast moving current density-- there's a negative charge excess because positrons can annihilate with atomic electrons). To detect this Askaryan emission, you need a dense dielectric material (if not dense, no target mass, if not dielectric, then RF won't propagate). Antarctica happens to be both the place you do long duration ballooning (due to all-day sunlight and favorable wind patterns that keep you over land) and the place with the most ice.
However, the events discussed here were produced by another channel. ANITA can also see RF emission from cosmic-ray extensive air showers (EAS). The RF emission here mostly comes from the splitting of charges in the showers by the Earth's magnetic field. Because in Antarctica, the magnetic field is approximately vertical, this produces horizontally polarized emission. Because ANITA is so high up (~40 km), EAS development from cosmic rays occurs below the payload, so the most common way for us to observe EAS's from cosmic rays is for the emission to bounce off the ice (because it's very forward-beamed). We can also see atmosphere-skimming showers that miss the ice entirely. As expected, the events that bounce off the ice have a polarity flip compared to the events that miss the ice.
The strange events discussed here look like EAS's from air showers, but the RF emission clearly points at the ice and there is no polarity flip from reflection, so the events look like very-energetic upward going air showers. There's no good way to explain upward going air showers in the Standard Model at these energies and observed angle (at lower energies or more grazing angles, tau neutrinos make it through the earth, which can decay to make upward-going air showers). So either there is something wrong with the measurement (we can't think of anything, but we're trying!), we got really unlucky with anthropogenic backgrounds (we think this is very unlikely), or there might be some new physics.
For this detection channel, there isn't too much special about Antarctica, just that we're on a balloon looking down so we can see stuff coming from below. The ice could potentially offer a slight enhancement compared to rock, but that's probably not so important. Other observatories looking for upward going showers from tau neutrinos (Pierre Auger) only look at very grazing incidence. There are proposals using fluorescence instead of radio emission (e.g. JEM-EUSO, and the SPB-EUSO balloon mission) that could do more or less the same thing.
Also a bit gobsmacked you can get Cerenkov radiation in air. On reflection I shouldn't be.
What about for the Askaryan channel?