How do horses eat in the game?

I was super confused too, killed a deer and couldn't eat it so thought "oh no, I just murdered for no reason."

Turns out you do still need to fight other animals. But instead of eating them afterwards you need to keep an eye out for the bush spawning right next to you. When you kill an animal it spawns a bush for you to eat, this is how you get your XP and level up.

You have to kill something, then walk over to where it dropped and eat it like you would as a normal animal, like a wolf. It will look like a bush.