Arrogance. If you read the story carefully, the reason why the second fire went out, and really the reason why the second fire had to be lit in the first place, is pure arrogance. A key theme of this story is how dangerous taking nature for granted really is as we are presented with an unnamed protagonist who tries to battle through the deep Alaskan winter and fails dreadfully, resulting in his death. If he had had more respect and understanding of nature and his place in it, he would not have died.
However, the second fire specifically goes out because the protagonist had made it underneath a spruce tree which was covered with snow. As he pulled out twigs to start the fire, the snow was disturbed until finally the snow that was on the top bough fell, causing a mini-avalanche as it hit the other lower-down branches and extinguishing the fire:
It grew like an avalanche, and it descended without warning upon the man and the fire, and the fire was blotted out!
Thus it was the lack of thinking and foresight of the man in placing the fire under the spruce tree rather than in the open that resulted in its becoming extinguished.