How do you stop your cat from bringing in live animals?

Never rub his nose in it. Or otherwise punish your dog.

In a pack hierarchy, a subordinate dog will bring his kill to the alpha to gain approval and to show they respect their leadership. The dog doesn't understand why its so gross to bring it inside, they are giving you a gift. They will get super confused if you become angry and put his nose in it. Your dog is doing something nice for you!

Take the carcass outside, set it on the ground, and praise your dog. Leave your dog with its prize for a little while (if he eats it, as long as its not rotting meat, it doesn't matter) to show that you accept the gift but you are "done eating" and it is the dog's turn to have its share. Most dogs probably won't eat it, they will just leave it there as a symbol of their accomplishments, and then you may go back and remove the carcass.