For a few reasons. First, bats have an extremely low rate of rabies, less than 0.5% of bats in the wild. They do have trouble transmitting the disease, bats are small and relatively fragile, most rabid animals that bite them will kill them, but not always.
I think the reasons bats are the most common transmitters of rabies are the following reasons:
-People have a general awareness of how rabid dogs and other such animals act, and stay away from these mammals when it is clear they have rabies, these animals quickly get reported to animal control and destroyed
-Most mammalian pets that can get rabies are vaccinated and don’t live in the wild, so have less opportunity to get the disease (if they aren’t vaccinated) or to transmit it.
-People have very little direct, hand-to-hand experience with bats. I have a bat house near my home that I put up because I like having swarms of bats around (no, seriously, they eat an obscene number of insects, the brown bats in my bat house eat something like 6,000 mosquitoes per bat per night.) Yet, I’ve never touched a bat, had a bat in my home, or had one get really close. Unlike birds bats tend to not be around during the day, and they don’t tend to walk around your porch like some birds do.
-Because bats are an oddity, when people get the chance to interact with one closely, they may not behave appropriately. A child may try to pick up a downed bat. Any bat that is on the ground during the day has something wrong with it, many such bats are the small portion of bats that have rabies.
-If a rat or mouse gets rabies, it’s still limited by the fact that they can’t fly, meaning even if they’re in the home, they’re unlikely to be able to bite you if you’re sleeping in a bed raised from the ground. Bats on the other hand, when they spread rabies, often do so when they bite someone in their sleep. The person may never even see the bat or realize they’ve been bitten. This ability to fly is probably why not every bat who is attacked by a rabid animal dies, even if wounded, as long as they can still fly away, they can flee most predators.