You could contend that the snowy, chilly environment has a direct relationship to the three men. None of the men seem to show much warmth of feeling towards one another or other people. Kenny playfully tries to run Tub over with his truck. Meanwhile, Frank appears to be having something of a predatory relationship with a teen babysitter.

You could also argue that their relationship to each other and the world has less to do with the cold and more to do with hunting. Kenny, Frank, and Tub are not only hunting animals, they’re hunting each other. They’re taking the hunter–prey mentality and applying it to people. Even Tub, who seems to be the least hunter-like of the group, shows his hunter-like mindset when he shoots Kenny.

I also think you could turn the question around and argue that the seemingly hostile environment is meant to conceal or mask the feelings these men actually have for each other.

You could contend that Frank, Tub, and Kenny treat each with such apparent apathy and disdain because they’re conforming to the norms of how men relate to one another. Maybe Kenny’s attempt to run over Tub or Frank’s teasing of Tub’s weight could be seen as a form of bonding.

More so, if the three didn’t possess some underlying connection, the personal disclosures from both Tub and Frank don’t really make sense. The sharing of intimate information about the babysitter and the weight seems to suggest just that: intimacy.

In some ways, this story is reminiscent of pieces written by Naturalism writers like Stephen Crane. A major component of those pieces is nature's uncaring attitude toward the plight of human struggles. Humans simply exist and shouldn't be given much more thought than that. We don't deserve nature's kindness. Readers can see that in this story in how nature never seems to help out the three hunters. They struggle to find deer, they struggle to move through the snow, and they struggle to get over, through, and around natural obstacles. There seems to be a cold indifference or even hostility from nature toward these three men. Those feelings fit quite well with the story because nature is essentially reflecting how the three men treat each other. The men might be friends, but readers can't help but think that maybe each man would be happier without the others. Kenny ridicules everybody, Frank intentionally feeds Tub's gluttony, and nobody seems to care that Kenny is wounded in the cold truck. The men are every bit as callous and indifferent as the story's natural setting.

Hunting is, arguably, an activity that requires a certain cold-bloodedness. In this story, Tobias Wolff capitalizes on that metaphor with the setting of a snowy and frozen environment, both in town and in the woods, and people who are equally cold-blooded in their interactions with one another.

The early characterization of Tub leads readers to believe that he is the unfortunate victim of a callous society: first by the driver who does not pick him up in the freezing weather, and then by two "friends" who ridicule and humiliate him for most of the day.  Even his nickname, "Tub" is not one he would choose himself; it has been chosen by others to shame him. However, as the story progresses, Tub changes from victim to participant in cold-blooded behavior when he shoots Kenny and then neglects to get medical attention for him.  

There is no warmth in the family whose dog Kenny kills, toward their pet, each other, or the strangers who ask for their help with Kenny's life-threatening injury.  Frank's plan to leave his wife and children for a teenage girl and the barbarity of Tub and Frank toward Kenny suggests that the world they inhabit is cold and filled with physical and moral dangers of which they show little awareness.

The cold, harsh environment relates to the meaning of the story of this existentialist work because of the way in which the snow, which has covered the landscape and made it impossible to navigate or to see any signposts, is representative of the way that any moral code has been covered over or dispensed with between these three friends. Note for example the following paragraph that makes reference to the way that the snow has covered up the landscape around them:

They left Spokane and drove deep into the country, running along black lines of fences. The snow let up, but still there was no edge to the land where it met the sky. Nothing moved in the chalky fields. The cold bleached their faces and made the stubble stand out on their cheeks and along their upper lips.

As the landscape has been covered up, both literally and metaphysically, the characters are free to operate without any moral code whatsoever. This is shown in particular with the relationship that builds up between Frank and Tub. Male bonding is shown to be taken to its absurd extreme. Frank and Tub are so pleased with the friendship that they are developing, and the freedom that they have to be honest with each other, that they view this burgeoning friendship as being more important than the life of Kenny, who is slowly dying in the back of the vehicle. The way that at the end Frank drives in the opposite direction of the hospital and they give away the blankets that were covering Kenny does not bode well for Kenny's future. The weather therefore signals to the reader that the characters are plunged in a world without any moral signposts, just as the literal signposts have been covered up by the snow. It is a world where anything goes, and absurdity rules.