Humans have the ability to empathize with other living beings, human or otherwise. It's one of the things that makes us human. What you are feeling is respect for the animal.
I'm not sure I feel sadness when I kill an animal, but I am at a loss to come up with a better word. I enjoy all aspects of hunting, but I don't actually enjoy watching them die. The transition from alive to dead is something I have never liked to witness, but it is a part of the process and to ignore it would be like burying your head in the sand. Whatever I am killing, whether it's shooting a sick horse, shooting the coyote that comes into the yard, or the woodchuck digging under the foundation, or trapping the mouse in the basement, I want to end its life as quickly and cleanly as possible. I hate to see things suffer, even as I am trying to kill them.
It's complicated and difficult to describe, but it's part of life. Things die so others can live. Every deer or game animal we kill means one less animal in a factory farm. The deer living a natural life, and then receiving a quick death from a responsible hunter, is about as good a death as a wild animal can expect. The natural ways they die are usually a lot less quick and clean.
The fact that we are aware of our own mortality means we are also aware of others, both human and non-human. This is a good thing IMO.