In short, Rainsford denies the existence of feelings in the animals he hunts because he is convinced that since animals allegedly have no reasoning, their lives have lesser value, and there is no guilt in hunting them.
In the first part of "The Most Dangerous Game," Whitney tells Rainsford that hunting might be sport for the hunter, but it is definitely not for the animals that get killed. Rainsford goes into defensive mode, saying, "Who cares how a jaguar feels?" When Whitney presses the matter, saying animals can at least feel fear, Rainsford dismisses his objections further by saying that some are meant to be hunters, and others, the hunted, so there's no point in worrying about it. He also genuinely believes that animals have no power of reasoning—at least not reasoning sophisticated enough for them to be upset over being the object of a hunt.
Rainsford's denial of feelings or understanding in the game he hunts links him, in a sense, with the villain, General Zaroff. Zaroff later repeats Rainsford's ethos that life is divided into two categories—hunter and hunted—only he takes this philosophy to its extreme, including humans even in the "hunted" side of the equation. When Rainsford reacts with horror, Zaroff claims he only hunts "the scum of the earth," which for him includes those who are lower-class and/or non-white. In this way, the story suggests that Rainsford's philosophy about the nature of life itself is also flawed, even dangerous. Whether or not his experience as "the hunted" changes Rainsford's mind about the animals he hunts is left ambiguous at the end.