How did the rats get the black death and not die?
They did die. After they were infected by the fleas that were
the original carriers, they got sick. They transmitted the virus
inadvertantly through the food they nibbled off of in humans'
pantries. Humans would eat this food and get infected with the
black plague. Towards the end of their lives, as the virus
progressed, the rats would go a tad loopy and bite humans,
therefore directly transferring the virus to humans. The rats then
died. Other creatures would feast on the rat's corpse and be
infected. The cycle continued.