When animals are killed, their carcass can be skinned before being cut for meat. You do this by using the "Hideworking" skill menu (ALT+h) to skin an animal, then butcher for meat. The quality of the acquired skin depends on how many and what kind of hits the animal has taken on its body. Piercing and slashing attacks destroy the skin the easiest so it might be wise to use blunt attacks to finish the animal off, if possible, or avoid targeting the body, instead aiming for the head for example. Damaged skins produce lower quality and lower value furs/leather.