Why does most rain fall over warm spot on the surface of the earth and not the cold spots?

It's not that simple, and not always true.
Deserts are the hottest places on earth and get little or no rain.
The poles and areas near the poles get tons of precipitation in the form of snow.
Aside from all that, land nearer the equator AND near to vast bodies of water does get plenty of rain. But the Sahara is near the equator and is totally dry, and the poles are surrounded by water and get tons of snow.
Warm air is able to hold more water than cold air.
There's a lot more to all this, including air patterns being different along the equator... that you can research on your own.