Well, I think the setting and context of the story is bound to add a certain element of fear and suspense: two expert hunters in a battle to the death played out on an isolated island... you get the picture. However, what the author does to tantalise us and instil further suspense into the story is during the actual hunt it becomes a battle of wits between the General and Rainsford. During the hunt there are three separate stages when Zaroff lets Rainsford go away and move on to be hunted again. Thus Connell denies us the final confrontation that we want and expect by putting it off for another few heartbreaking moments.
Consider the first example of this:
Rainsford held his breath. The general's eyes had left the ground and were travelling inch by inch up the tree. Rainsford froze there, every muscle tensed for a spring. But the sharp eyes of the hunter stopped before they reached the limb where Rainsford lay; a smile spread over his brown face. Very deliberately he blew a smoke ring into the air; then he turned his back on the tree and walked carelessly away, back along the trail he had come.
It is clear that Zaroff knew of Rainsford's presence on the tree, but in the interests of securing an "interesting" game, he decides not to finish the hunt straight away, rather allowing Rainsford further chances to test his skill against the General's hunting prowess. As the author describes it, the General always returns quickly:
The cat was coming again to play with the mouse.
This playing of the prey clearly is an appropriate metaphor which itself builds suspense - the cat almost always wins, but will the mouse triumph? We are compelled to read on to the gripping finale to find out.