Your question identifies what this short story is so famous for, and why it is so successful. One of the ways that writers create suspense is through foreshadowing, the use of clues that hint at later events in the story. Foreshadowing makes you curious, even anxious, to know what will happen next.
For me, one of the first pieces of foreshadowing that occurs in the story is in the discussion between Rainsford and Whitney. Note what they say:
"The best sport in the world," agreed Rainsford.
"For the hunter," amended Whitney. "Not for the jaguar."
"Don't talk rot, Whitney," said Rainsford. "You're a big game hunter, not a philosopher. Who cares how a jaguar feels?"
Of course, this is ironic as Rainsford himself is going to discover how the jaguar feels as he becomes the hunted rather than the hunter.
The second piece of foreshadowing I will focus on comes very close to the first. The evil reputation that the island has clearly forebodes some kind of ill, as we will later discover. Note how this is introduced:
"Yes, even that though-minded old Swede, who'd go up to the devil himself and ask him for a light. Those fishy blue eyes held a look I never saw there before. All I could get out of him was: 'This place has an evil name among seafaring men, sir.'
This clearly hints at some kind of danger that will feature in the rest of the story.
Clearly another highly significant piece of foreshadowing occurs once Rainsford is on the island and he sees the evidence of a hunt, but he is not able to work out what kind of animal was hunted:
Some wounded thing, by the evidence a large animal, had thrashed about in the underbrush; the jungle weeds were crushed down and the moss was lacerated; one patch of weeds was stained crimson.
Of course, later on Rainsford will no precisely the identity of the species of animal that was hunted.
So there you are - three examples of foreshadowing which arguably help to make this a unique story of suspense story and one which keeps the reader engaged.
Connell usese foreshadowing in several places throughout "The Most Dangerous Game." This use of a literary device that encourages the reader to feel a sense of anticipation about what will happen to Rainsford is foreshadowing.
In the beginning of the story, Rainsford and Whitney discuss the blackness of the night and the boat's close proximity to Ship-Trap Island. Whitney's insistance that sailors have a "curious dread of the place" leads the reader to wonder why they would feel that way. The reader begins to anticipate the possibilities of that night and location. Connell's use of phrases such as "thick warm blackness" and "moonless Carribean night" turn the reader's mind toward eerie suggestions. When the discussion turns to hunting, the idea of danger and death come into play.
Later in the story, General Zaroff states that his mind "is an analytical mind...Doubtless that is why I enjoy the problems of the chase." Zaroff also informs Rainsford that
...hunting had ceased to be what you call 'a sporting proposition.' It had become too easy. I always got my quarry. There is no greater bore than perfection...Instinct is no match for reason...
This information causes the reader to ask himself how Zaroff dealt with his problem of boredom. Suspense is built as the reader realizes that they prey can only be human, which Zaroff soon admits, and that Rainsford must either become the hunted or join Zaroff.