Finding himself soaking wet and with the temperature dropping, the man tries to light a fire to dry out. His first attempt succeeds, as he works calmly and slowly, but the fire is put out by a clump of snow falling from a tree. At that point, the extreme cold has given him frostbite in his fingers, and he has trouble manipulating the matches and fuel for the fire. 

He picked [the match] up in his teeth and scratched it on his leg. Twenty times he scratched before he succeeded in lighting it. As it flamed, he held it with his teeth to the birch bark. But the burning brimstone went up his nostrils and into his lungs, causing him to cough spasmodically. The match fell into the snow and went out.
(London, "To Build a Fire," eNotes eText)

Every minute he does not build a fire, his frostbite gets worse. Finally he tries lighting all his matches at once; another mistake, as he succeeds in lighting some tinder but then is unable to form it into a larger fire because of his frostbite. He could have avoided the snow by building the fire in the open, but was rushed to thaw his feet for fear of losing toes. At the end, he fails because he did not prepare properly, and because he only carried one source of fire instead of two or more.