succeed in a hazardous undertaking for someone else's benefit.
This expression refers to the fable of a monkey using a cat's paw (or in some versions : a dog's paw) to rake out roasting chestnuts from a fire. Car's paw is sometimes used as a term for someone who is used by another person as a tool or stooge.
Related Idioms :
an old chestnut
a joke, story or subject that has become tedious and boring as a result of its age and constant repetition.
The most likely source for this sense of chestnut is in the following exchange between two characters, Zavior and Pablo, in William Dimond's play Broken Sword (1816) : ZAVIOR... When suddenly from the thick boughs of a cork tree - PABLO. (Jumping up) A chestnut, Captain, a chestnut... Captain, this is the twenty-seventh time I have heard you relate this story and you invariably said, a chestnut, until now.