"The Snows of Kilimanjaro" is a frustrating story about a frustrated man, who has misspent his life and wasted his talents, and now finds himself dying of slow blood-poisoning on a safari far away from home. He spends most of the story arguing with his wife, Helen, a patient, long-suffering woman, whose main offence seems to be not living up to Harry's ideals. He hates the way he treats her, as he hates the way he has behaved in all arenas of his life, and he regards his approaching death with a kind of bitter relief: here, at last, is an end to his confusion and suffering, and also to the suffering he's inflicted on others throughout his life. Helen urges him to remain cheerful, and insists that help will arrive soon, but Harry shrugs her off. As the night draws in, his infection worsens, and he begins to have oppressive premonitions of his death, which fill him with fear.
The passage you have asked about describes the final day of the couple's ill-fated safari. Harry, having spent a restless night slipping in and out of delirium, wakes to the sound of a plane overhead. This is the rescue plane which has come to airlift him to a hospital, where his gangrenous leg can be treated. Helen's optimism is vindicated by the plane's arrival, and that of the cheerful pilot, Compton, whose presence relieves Harry's feelings of dread. Serving boys help load Harry into the tiny plane, and he and Compton fly off towards civilization. The landscape below recedes into the distance, the plane passes through clouds and a sudden rainstorm, and then Kilimanjaro suddenly appears before them, and Harry realizes "that there [is] where he [is] going." As Harry has this realization, the point of view shifts back to Helen in the encampment, where it is still night. The sound of a hyena wakes her, and going to check on Harry, she discovers him dead.
There are a few interesting things about this climax, the most notable of which is the changing point of view. It is the first time we see things through Helen's eyes, and what we see is that Harry, far from being rescued, has succumbed to his infection. The hyena that wakes her with its "strange, human, almost crying sound" is the same hyena that Harry had been watching earlier in the evening, which he linked with his premonitions of death:
"[...] just then it occurred to him that he was going to die.
It came with a rush; not as a rush of water nor of wind; but of a sudden, evil-smelling emptiness and the odd thing was that the hyena slipped lightly along the edge of it."
Harry so strongly associates this hyena with death that he likens it to a kind of Grim Reaper, saying to Helen:
"Never believe any of that about a scythe and a skull," he told her. "It can be two bicycle policemen as easily, or be a bird. Or it can have a wide snout like a hyena."
When the viewpoint shifts to Helen's perspective, the hyena is still present, and making a very odd sound, but Helen was external to Harry's convoluted thought-process and therefore only hears a hyena, rather than anything supernatural. Harry's death shocks and saddens her, and she is insensible to anything outside of the dead man before her and the sound of her heart beating in her ears.
In this moment, as Helen kneels before Harry's cot, the reader can see how much she loves him. Harry has spent much of the story disparaging Helen, denying her intelligence, ability, and affections because he himself is so bitter and unhappy that he cannot accept her love. And yet when we see from Helen's point of view, we feel her panic and anguish at the thought of losing her husband:
"Then she said, 'Harry, Harry!' Then her voice rising, 'Harry! Please. Oh Harry!'"
And Harry, up in the sky flying towards the mountain, is for the first time, free from all the anger and confusion that has dogged him his whole life. As the plane lifts over the landscape he can see the beauty in it, and the patterns that make sense of the whole. It is only at the point of dying that Harry has had the time and inclination to review his life, and in the past few days, he has found it disappointing in every regard. Now, in the plane bound for the hospital where he will be healed of his sickness, he finally feels peace; he is willing to heal and to go on living. To realize that he is, in fact, dying, does not fill Harry with fear the way the specter of the hyena did; instead, he is filled with awe:
"[...] there, ahead, all he could see, as wide as all the world, great, high, and unbelievably white in the sun, was the square top of Kilimanjaro. And then he knew that there was where he was going."
As Helen's perspective on Harry's death shrinks the world to the sound of her heartbeat, Harry's own perspective is suddenly freed from all the "surly bonds of earth," as the plane passes through a storm and emerges in front of the shining white face of the mountain. This may be understood as the moment of Harry's dying, when his soul breaks its link with his body and returns to "Ngaje Ngai, the House of God."