Primarilly the enjoyment of the hunt itself. Getting out in nature, seeing more wildlife in one day than most people see in their entire lives. But also the challenge of matching wits and skill with animals or birds that been evolving for millions of years to learn better ways of avoiding predators... including man. The kill is of secondary importance to whether I have a great day hunting, but I DO get enjoyment out of killing my own food... as I get enjoyment out of eating vegetables I've raised myself. Feeling of self-sufficiency... feeling of getting back to accepting the challenge of taking on nature one-on-one. Hunting is NOT just shooting or killing... it's a LOT more challenging than you'd expect. Takes a lot of preparation, skills learned over time. You have to learn to think about what the animal or bird is thinking... what do you have to do to be in the right place to get a shot.
And how do I feel? Satisfied... and even a better feeling when I watch my son hunting with me shoot a duck or goose.