How do whales hunt and eat prey without swallowing gallons of water?

They filter the water. If you've ever seen a baleen whale, they have a big floppy sack under their jaw, called a baleen, which works like a giant tea strainer. The whale takes a mouthful of water and then squirts it out again through the baleen, trapping all the plankton they want to eat in there so they swallow just plankton and not mouthfuls of water.