What do the percentages under rain mean?

Chance of precipitation.

Rain percentage is a standard measure in meteorology. It is a compound value that takes into consideration the probability of rain falling anywhere in the forecast area adjusted by the size of the rain source. So it is the probability of rain falling specifically on you if you stay in one location. For example, if the radar image shows an positively identified approaching storm that is half the size of the forecast area the rain percentage would be 50% for any location in the forecast area. That’s a gross oversimplification, but I hope it gets the idea across.