The cat has claws on its legs and it chases down and grabs the rat and bites its neck = dead rat. I haven't witnessed this, but I think that's what's happening. Be nice to the cat, it's doing you a service.
I've made friends with a feral cat that's adopted me recently. I started putting out water and a little cat food a few days ago to solicit its favor in hanging around. I'm hopeful it will control rodents and most particularly possibly squirrels, which reduced my Japanese pumpkin crop by about 10% last summer. I've heard that only certain cats will mess with and control squirrels, though, so it's probably iffy. This cat isn't too big, I suspect it's a female.