A game of fetch relies upon a cat's natural urge to chase and retrieve prey for its young kittens and to carry things in its mouth. Cats are natural-born hunters. They have instincts which make running play irresistible and encourage them to carry prey or toys.
Human owners can apply those instincts to a fun game of fetch if they train their cat when it is young. Some cats are predisposed to fetch regardless of being trained, especially cats of Siamese heritage. They spontaneously begin to fetch for their owner.
A cat's body is designed for carrying. Mother cats carry their kittens in their mouths and also bring back small prey for a hungry kitten that needs lessons on hunting. Their lips are able to curl back loosely to hold objects in place, and they can modulate their bite force to lift and support things gently.
Cats that do not fetch often display the carrying urge by bringing their owners dead or injured prey, the way a mother cat would for her young. Cats teach hunting to their young by having their kittens provide the killing blow to a prey animal that has been severely injured by the adult cat.