A big thing is that Bats have a really weird metabolism, during their day it can dip down to 50 Fahrenheit and then go up to 104 F at night when they are active. This can mean they end up carrying a lot of diseases but not dying/showing symptoms of them. They also have very strong DNA repair compared to other mammals. Then in addition many bats are very social and sleep in big groups, which means the disease can spread throughout the bat population.

Edit: Finally and relevantly they can come in close contact with people by coming into our homes, or people going into theirs. This can let the disease cross over.