Hunting is entirely natural for cats, and takes place even when they are well fed at home. They enjoy the hunt, stalking patiently and carefully until they are close enough to pounce.
If your cat returns with her kill and presents you with a mouse or a dragonfly, she wants you to congratulate her on her hunting prowess. There is really no point in punishing your cat for hunting. It is a part of her nature that is so deeply rooted that to try and eliminate it may cause her a lot of confusion.
One solution is to put a bell on her collar so that the birds and other likely victims can hear her coming. However, many cats have learned how to stalk without making a sound, despite the bell. If you do put a bell on her collar, make sure the collar has an emergency release so she can escape if it gets caught on some object. Most pets stores carry this type of collar.
Finally, if your cat likes to hunt outdoors, it's important to treat her regularly for worms, as she can get parasites through eating her prey.