Ernest Hemingway became famous in part for his introduction of a clean, terse writing style, approximate to a journalist's approach of letting the "facts speak for themselves," and "The Snows of Kilimanjaro" is a case in point. The story is modernist in terms of both form and content. The narrative employs innovative techniques that draw in the reader, making him or her work to realize the meaning of the piece, even as the protagonist is pondering the meaning of his (waning) life. Further, "Kilimanjaro" depicts themes - sex, class hostility, alcoholism, graphic recounts of war - that at this point in literary history (1936) were rarely dealt with as directly. And he does so in economical yet devastating prose.
Modernist technique opens the story; it begins abruptly with dialogue rather than any attempt at a descriptive introduction or presentation of the plot. We can't tell who is speaking, to whom or what the issues are. We get a sense there are vultures hovering, but they are not indicated by name, and we don't know what their prey is. The introductory quote about the mountain's history and the mysterious dead leopard is tantalizing, but unexplained: what was the leopard doing at that height? What does it have to do with the story? The effect is to make the reader a bit of a sleuth; s/he has to read closely to ferret out the details of the story, to learn that the buzzards loom about Harry as he is dying from gangrene to the leg, attended by his rich, privileged wife and the African natives that accompany them on safari. We are confused about what will happen to Harry, even as he is himself facing the unknown; we take in the enormity of his situation as he begins to realize it himself.
This formal technique and innovative presentation of point of view are elements of Modernism.
In addition, the subject matter of the narrative is modern: it is graphic; sexually explicit; psychologically penetrating; and includes social and class critique contemporary for its time. The story acts as a retrospective of Harry's life as it begins to dawn on him that he is dying. Harry recounts in explicit detail his experiences of war and of love. He tells of "urine-soaked roads" and the soldiers he skied with in Austria that he would eventually kill. We do not get the sense he is an unquestioning proponent of the war.
Harry is angry about a lot of the history of his life, and much of his anger seems to center, in psychological terms, on his wife. She is a "rich bitch." Because of her, he will never realize his potential or cover the subjects he intended to as a writer. She has "broken his balls," so to speak, sexually and as a man -- because she has the money in the family, she takes power from him. Increasingly easy on the cushion of privilege and wealth she provides, he gets nothing done, fritters away the time on frivolous ventures like the current safari and now has encountered an accident that will end his life. Like the gangrene in his leg, he is rotting away, and as much as he blames himself, he also directly blames his wife. Through Harry's perspective, Hemingway presents the classic Freudian trope of the castrating female.
Thus in the employment of formal technique, subject matter and social commentary, "The Snows of Kilimanjaro" is an excellent example of Modernism.