How does the man's attitude begin to change when he realizes death is approaching?

I think that the man realizes death is coming for him when the snow falls on him and the fire and extinguishes the life saving heat. At this point in the story, he remains very calm. He knows that he is in a life or death situation; however, he also knows that he has the tools to relight the fire if he is careful. Unfortunately, all of the matches get burned up at one time. It's soon after this that the man probably realizes that he is going to die. He moves from being calm to panicked quite quickly, and he starts to run around the place "wildly":

But the thought returned and continued, until he could picture his body totally frozen. This was too much, and again he ran wildly along the trail.

The man eventually falls down in the snow, and he realizes that he is acting foolish. He knows that he is going to die, but he doesn't want to die in a panic. He wants to meet death calmly:

It was his last moment of fear. When he had recovered his breath and his control, he sat and thought about meeting death with dignity. However, the idea did not come to him in exactly this manner. His idea was that he had been acting like a fool. He had been running around like a chicken with its head cut off. He was certain to freeze in his present circumstances, and he should accept it calmly.

The man sits down and begins separating his mind from his physical body, and he admits that the old man of Sulpher Creek was right.

The attitude of the man towards death veers between denial and acceptance. This a broad theme that applies to most people throughout their lives. Most of us go through life without thinking of death. The very thought of it is just way too unpleasant. It's only at certain moments in our lives, such as the death of a loved one, for example, that we start to contemplate our own mortality.
The man in "To Build a Fire" finds himself thrown into just such a situation. His natural instinct for survival is initially quite strong and manifests itself in different ways. For instance, when the matches start to fail and he can't light a fire, he starts to think of other ways to keep his hands warm. Maybe I should kill the dog and warm my hands inside it, he thinks.
But eventually, the man realizes that the game is up. Once he finally understands that there is no longer any hope of survival, he accepts his fate. It's now time for him to face his imminent death with dignity, instead of frantically running around in the snow, trying to avoid the inevitable. So he sits there in the snow, calmly waiting for the end. His mind starts to separate from his body, as if he's looking at himself as another person; he imagines what the boys will see the next day when they find his corpse in the snow.