The fleas carried the actual disease, and got about on the rats. They would feast on whatever warm blooded creature was on hand, and often these were the tenants of whatever house the rats were infesting.
This was an unfortunate byproduct of superstitions in the middle ages. Cats were made the scapegoats for many ills. They were linked with witchcraft, the devil, and crib death, among other things. As cats were tortured, killed, or driven out of areas, the rat populations skyrocketed and the fleas increased in turn.
It didn't take long for the plague to take hold.
It still pops up now and then, primarily in South America.
(Did a study on it for high school)