My cat does this too. I think it's very cute, but I'm having trouble figuring out how to deal with it logistically. The problem is compounded by the fact that my other cat frequently pulls her food out of the dish onto the floor (yes, I'm still using dry food, boo hiss).
Any recommendations? I tried one of those little mats, but my cat seemed to just flat out mock me for that. The burying cat would pull some or all of the mat over the food, sometimes spilling the water dish.
I tried putting the dishes onto a tiled surface, but the burier still has fully functional front claws and so he would scuff the floor. And the other cat would leave little food remnants on the floor.
My current solution, I'm not recommending it, is to put a cheap rug underneath and weigh it down. Now when he tries to bury the food, at worst he just tears up a little bit of the carpet. The concern now is that the other cat is eating off of a questionably unclean rug. I do clean it sometimes but apparently not often enough.
My kitties do it too, but only with wet food and if they don't finish. They'll mark all around their plates with their paws and I always took it as a sign of, "This is mine. Don't touch it." They'll usually come back and finish eating 10-15 minutes later.