Reading the letter is an attempt to lighten the mood in the bunk house, but it is ineffective. Ironically, it would probably be more comfortable if they weren't trying so hard. The fact that they do try to provide a distraction, though, shows compassion on a ranch where such things are rare.
The fact that the men remember Bill Tanner is interesting, too. Most men, it seems, come and go anonymously--disappearing from existence. The fact that he has reached a kind of celebrity status because of his letter speaks to how lonely and disconnected these men really are. It is another subtle reminder of the plight everyone on the ranch is faced with. What will they leave behind? Who will know that they existed?
The Letter to the Editor is a conversation between a bunkhand and Slim about Bill Tanner, a former employee. The conversation is mundane, with no direct relevance to the story itself, and is a contrast to the tension in the room as Carlson has taken Candy's dog up the hill to put him out of his misery.
Everyone is waiting for the sound of a gunshot. The mundanity fills the space and is in such contrast to the sadness of the event that it actually accentuates it, draws the reader's attention to it.