The text plays on the double meaning of 'bury the cat'. It could mean to literally bury a deceased cat, which is the expected meaning, but the humor arises from the revelation that the cat was actually moving, implying a struggle to keep the cat still for a non-lethal reason, such as a bath or a nap. This creates a darkly comic effect.
Took an hour to bury the cat . Silly thing kept moving . . .