Dynamic characters undergo changes during the course of a story. Rainsford undergoes changes on two levels. He is able to change and adapt physically during the hunt with Zaroff thus ensuring his survival. More importantly, however, he undergoes an internal change during the hunt. Before the hunt, Rainsford was himself a big game hunter with little regard or concern for the animals he hunted, but with himself becoming the hunted, he has an "epiphany" of what hunting another living creature involves thus perhaps changing his perspective somewhat.
He is a dynamic character because he is able to change and adapt as the situation changes. At first, he is a hunter-- then he becomes the hunted. Lastly, he defeats the hunter and the possibility is left open that he himself could take the place of Zaroff. His ability to think and act makes him dynamic, and the fact that his actions change based on the situation makes him highly dynamic.