One of the things that impresses me about hunting is that it is such a natural activity. What I mean by this is something phenemenological, it is a certain sort of feeling I have while hunting.
Let me try to put a finer point on that. I have in the past tried to go backpacking "purely recreationally". That is, I've gone to the mountains as a "pure recreational hiker" might--to see them, view the wildlife, and spend time there (or otherwise "commune with nature"); but without interacting with the mountains the way a hunter would. That is, without evaluating the hunting (or fishing) potential of the area. I won't try to say here what that means (though you might take a peak at some of my thoughts in this post).
The result was an experience of the wilderness that was somehow forced or abbreviated. It is not that I can't or won't appreciate the woods from a nonutilitarian perspective. It's just that I felt as if there was something further I ought to be doing; something I felt compelled to do, but from which I was refraining. [Analogy: think about downhill skiing in a situation where you are skiing below your skill level. You constantly feel like you ought to be pointing your skis downhill and heading off, but you are fighting that tendency. Your actions don't feel natural] And this, I think, is part of what the "feeling of naturalness" comes to. It is the feeling associated with performing an action that is neither forced, nor artificially restrained.
Now the thing is that most of this behavior is just a refined version of what most people find it natural to do in the woods. For instance, take a six year old to the mountains. It is usually pretty easy to get them interested in an elk wallow or deer scat; much harder to get them interested in granite formations (except to climb on). So the sense in which hunting feels natural correlates with the sense in which hunting comes naturally.