Some cats will hunt and eat the mouse they catch. This is a learned behavior, so not all cats eat what they kill.
Hunting is a natural behavior for cats. They will instinctively catch mice. However, eating the prey is a learned behavior. A mother cat will catch a mouse and bring it to her kittens. She will eat it and allow them to have the next one. After a while, she will bring live prey so the kittens can learn to kill. This is also why cats bring prey to their human. The cat is attempting to show the human how to hunt.