The mechanics are that the skin gets more damaged the more damage there is to the body of the animal. Thus, once you've cornered, caught (in a trap), or rendered the animal unconscious you should aim for the head exclusively, as attacks that hit any other body part (including the neck) is causing damage to the skin.
A large animal can take a fair number of hits to various body parts before the skin gets damaged, while a small animal like an ermine gets the skin damaged by an instantly killing arrow through the neck.
The above is not intended as a defense of the current system where feather retrieval is correlated to skin retrieval, but rather to help understanding of how the current system works.
It can also be noted that there aren't that many uses for feathers, and you'll probably get more than you have any use for rather quickly, but that doesn't help someone who's just started out.