In the novel Hatchet, Brian attempts to survive in the wilderness with very few resources, in a mostly untraveled and desperately cold part of Canada. Anyone who has been a survivalist knows that fire is one of the most vital resources when attempting to survive or find rescue.
Brian feels that fire is his friend because it will keep him warm and allow him to cook his food to keep him alive during his ordeal. Without it, he would quickly freeze to death, and, if not, would likely starve because his sources of food (mostly fish and small animals) would be inedible without cooking. Beyond the simple idea of keeping himself safe and alive through heat, fire also provides light and smoke, which act to signal potential rescuers from a great distance. He misses an opportunity to signal a passing plane with his flame, but it would have made an excellent chance at salvation.