Even more accurately, the neutrinos themselves cannot create Cherenkov radiation, as they don’t have electric charge. But every once in a while a neutrino interacts with the matter in the ice, and thus creates secondary particles, some of which will (a) have charge and (b) move faster than the local speed of light in the ice, and this will create Cherenkov radiation.

The main reason it is so large is simply that it can detect more neutrinos than smaller ones, but the construction and distance between sensors and thus its sensitivity to certain neutrino energies is in a way also a function of its size.