Question

why does the leeward side of a mountain not get much rainfall?

Answer 1

This is because the prevailing winds come up and the cool air tries to hold that much moisture. The dry air moves on to the leeward side (the side of a mountain that is not facing prevailing winds) and stays  there, usually causing deserts to form.

Answer 2

Answer:

The leeward side typically has a dry, warm climate.

This is because by the time air rises up the windward side and reaches the summit, it has already stripped of the majority of its moisture. As this already dry air descends down the lee, it warms and expands (a process known as adiabatic warming), which causes clouds to dissipate and further reduces the possibility of precipitation. This occurrence is known as the rain shadow effect. It is the reason why locations at the base of a mountain lee tend to be some of the driest places on Earth.