Brian isn't the narrator of the book—we know. But Hatchet is absolutely his story. And the fact that we see things from his point of view is clearly reflected in the book's tone. There's very little distance between Brian's thoughts, feelings, and perceptions and the attitude of the narrator.
Take a look at this passage from late in the book, when Brian is trying to escape the crazy moose who's been attacking him:
He started to move, ever so slowly; her head turned and her back hair went up—like the hair on an angry dog—and he stopped, took a slow breath, the hair went down and she ate. Move, hair up, stop, hair down, move, hair up—a half foot at a time until he was at the edge of the water. (16.19)
The narrator doesn't comment on what Brian does or thinks: he just reports it in a straightforward manner, without any sense of judgment or evaluation. He tells us what's going down as it's experienced by Brian.
Because of this, and because Brian's situation is often desperate and almost always stressful, the overall feeling of the book is super intense. Through some specific stylistic tricks (see the "Writing Style" section for more on this), Paulsen's writing mirrors Brian's keyed-up emotional state—which, by the way, mirrors our own as we're reading.