Candy’s dog is old and smelly, and for this reason Carlson shoots it.
Candy is the old swamper, and he has a very old dog. The dog is considered too old and smelly, and Carlson offers to shoot it. Candy can just get a new puppy to replace it from Slim. The shooting of the dog is both symbolic and foreshadowing. It sets a precedent of putting living things out of their misery when they are beyond hope.
And at his heels there walked a dragfooted sheepdog, gray of muzzle, and with pale, blind old eyes. The dog struggled lamely to the side of the room and lay down, grunting softly to himself and licking his grizzled, moth-eaten coat. (Ch. 2)
Because the dog is so smelly, the other men are tired of being around it. Since Candy is sentimental and attached to the dog, Carlson offers to shoot it. Candy does not want to at first because he has had the dog so long. However, Carlson wins out. He offers up shooting the dog as a charity, in its benefit. He wants to put the dog out of its misery because it is so old and in pain.
"That stink hangs around even after he's gone." He walked over with his heavy-legged stride and looked down at the dog. "Got no teeth," he said. "He's all stiff with rheumatism. He ain't no good to you, Candy. An' he ain't no good to himself. Why'n't you shoot him, Candy?" (Ch. 3)
Still, the shooting of the dog is a difficult thing for Candy. A new puppy will not replace the dog he has had since it was a puppy. Carlson reminds him that the old dog just “suffers hisself all the time” and Candy gives in “hopelessly” because he feels he has no choice.
The shooting of the dog foreshadows Lennie’s death later on, when George shoots him so that he will not be caught. Lennie has accidentally killed a woman, but he doesn’t understand the situation he is in. George knows this, and takes a page from Carlson’s book. He shoots Lennie to put him out of his misery. He is doing his last act as Lennie's protector and friend.