Rainsford does not believe animals and people are the same. Here he claims that he is a hunter and that if General Zaroff hunts people, he is a murderer. This implies that Rainsford does not feel as if killing animals matters or that they feel anything at all the way humans do. He believes that a kill for sport can only be considered murder if it is a human being killed.
Earlier on in the short story, he also mentions directly to Whitney that he believes animals feel differently than humans do, and that their lives do not matter in the way his and Whitney's do.