Peace Like a River, by Leif Enger, is the story of the Land family: Jeremiah and his three children, Davy, Reuben, and Swede. They live in North Dakota, at least when the story begins, and the family goes hunting one cold day. As you suggest in your question, these scenes do have a kind of universal appeal for many readers, even those who have never been hunting.
The first example of this universality is Enger's use of humor to create a familiar scene. The youngest, Swede, is determined not to let a wounded goose get away, so she chases after it,
yellow hair bouncing behind her stocking cap. Downfield, though, the goose seemed to have recovered its wits. It stood upright, taking stock, its head so high and perky I feared it might take off and fly after all. When it saw Swede coming it turned and sprinted away.
Swede is persistent and the goose tries to "poke its beak" at her. It is a comical sight, and people who have never been hunting, have never been to North Dakota, and have never had siblings can relate to this image because we have all found humor in the animal world.
Another way Enger makes these scenes universal is by using language which allows readers to connect to the experience. The Lands went hunting early in the morning and will go again in the afternoon. After breakfast, they each go to their own rooms to nap. "That's how goose-hunting is--you rise early and do the cold, thrilling work; then come in and eat; then fatigue sneaks up on you and knocks you flat." This is a common experience for most Americans and is certainly not exclusive to hunting (think Thanksgiving).
Finally, Enger uses the camaraderie of a shared goal to create a universal experience for his readers. The hunters in this case are all family, but nearly everyone has worked together with others on a common task. Reuben narrates it this way:
Dad and Swede lay on their elbows side by side, the two of them whispering under a swath [of barley].... We squirmed on our bellies up a shallow rise beyond which a few dozen honkers were feeding on stubbled wheat. This time there was no whispering among us; the light was almost gone and though we supposed the geese were close we couldn't hear them and had to crawl in faith that they were there at all. It was a very serious crawl, even though Swede on my left was pretending to be a wild Sioux brave creeping up on some heedless cowpoke.
The combination of humor, effective language, and a shared goal ensures that even non-hunting readers can relate to the Lands' hunting experience.