KH is alkalinity, or the buffering capacity of your water. The main reason aquarists test for this is to determine how stable pH will be in your tank.
Buffering capacity is an indication of a stable pH, in that a low (poor) buffering capacity means pH can drop drastically with the introduction of acidic compounds (fishy waste, decaying organics like driftwood, peat, etc.). We like to see greater buffering capacity, or higher
KH, because that means it takes a whole lot more acid to bring down the pH, thus making pH a more stable value in your tank, which is a good thing for your fish, since any tank over time will become more acidic as fish waste collects in the nooks and crannies.
KH should not be confused with hardness of your water, which is
GH or general hardness, and is a very different thing. It is simpler and more accurate to think of
KH as akalinity or buffering capacity, and link it to pH stability.