From the moment the man first takes off his mittens, you realize that his hands are going to be the star of the show. He tries to help his dog get ice off its feet, but is "astonished at the swift numbness" that strikes his fingers (13). At this early point in the story, his confused reaction shows you that he's not at all prepared to be traveling in such cold, and his hands will be the first to pay the price.
When his second fire gets put out by the snow falling from the spruce tree, the man recognizes that he's in deep trouble, and his failure to build a new fire is caused completely by the stiffening of his frozen fingers: "When he touched a twig, he had to look and see whether or not he had hold of it. The wires were pretty well down between him and his finger ends" (21). This image of fallen telephone wires helps convey just how drastically the harsh weather has shut down a vital part of the man's body that connects him to the world, just as a snowstorm would bring down telephone wires.
Much like the fire, the man's hands can mean the difference between life and death. By losing control of his hands, he is also unable to kill his dog and use its body to (wait for it…) warm his hands. Hands act as a sign of the man's prowess, his power. They are so important that he is even happy to feel pain in them, as long as he feels something: "After a time he was aware of […] a stinging ache that was excruciating, but which the man hailed with satisfaction" (27).
Ultimately, his hands and his fire meet in one final image of desperation, as he plunges his hands directly into burning flames: "His flesh was burning. He could smell it […] And still he endured, holding the flame of the matches clumsily to the bark that would not light readily because his own burning hands were in the way, absorbing most of the flame" (29). The pain is brutal, but the man understands that it's a matter of life and death, and holds them to the fire for as long as he can. Yet in the end, he loses both his hands and his life. In the most morbid of senses then, you could even say the man's life is in his hands. Just don't go stealing that phrase for the title of your next essay.