Sanger Rainsford thinks of himself as a "hunter", meaning someone who enjoys killing animals for sport. Unlike early hunter-gatherer societies or even contemporary poor rural families in the United States for whom an annual deer hunt can be the difference between adequate nutrition and need to rely on food banks or welfare, Rainsford is a sport hunter, killing for trophies and the thrill of the hunt, rather than just for food. He thinks of hunting as a sport or game, and has no sympathy for the animals he tracks and kills.
As Rainsford discovers that Zaroff hunts human beings, he makes a distinction between "murder," which is killing humans and "hunting", which involves killing animals. As the story progresses, and Rainsford eventually kills Zaroff, we find a distinction between "killing in self defense" and "murder." However, one of the main points made by the story is that rather than there being a clear distinction, as Rainsford initially imagines, in fact in the story we see a continuum between the two.