I'm at a friend's place and the cat keeps bringing in dead (or half-dead) animals into the house. It's my understanding that cats think we are big, helpless kittens that don't know how to hunt. Hence, they think they are doing us a favour.
It seems like a few mice actually escaped and found refuge in some walls in the house, so these "presents" are actually more than just annoying (and smelly if the dead animal ends up behind the couch).
There's a lot of wild cats that live across europe, which is why the advise you get in the states is different because over there, cats are more of an invasive species which its prey haven't evolved with.
We have many species, but here's one example which has been around for 170,000 years: European Wildcat