It comes from the behavior of opossums, which, when threatened, will sometimes go limp, roll over, and appear dead. This often discourages predators, allowing the opossum to escape.
Here are some examples of how "playing possum" is used:
* "The criminal was playing possum, hoping the police would think he was dead and leave him alone."
* "The child played possum when his mother asked him to clean his room, hoping she would forget about it."
Essentially, "playing possum" is a way to avoid a confrontation or unpleasant situation by pretending to be something you're not.