The short answer to your question is that most of them die. Uncle Billy runs off with the mules and supplies the first night, and Tom Simson walks back to Poker Flat for help on a pair of improvised snow shoes. Everyone else either starves to death or (in the case of Oakhurst) commits suicide.
In a larger sense, what “happens” is that this group of outcasts forms a kind of community. Even though the travelers have been run out of town as undesirables, when they realize their predicament, they band together and care for each other. This is particularly true of Mr. Oakhurst, the detached gambler. He knows from the beginning that the decision of the group to camp in the mountains is a mistake that likely will cost them their lives, yet he does not force them on, nor does he abandon them as Uncle Billy does. It’s as if he decides to embrace his better instincts. As his companions build a fire and soothe themselves with drink, Oakhurst “gazed at his recumbent fellow exiles, [and] the loneliness begotten of his pariah trade, his habits of life, his very vices, for the first time seriously oppressed him.” Rather than “playing the game out” and pushing on, Oakhurst decides that perhaps staying with this group is a better “game.” At any rate, his decision and his care for his companions turns the snowy mountain hut into a much more upright community than the town that had rejected them.