This dog is described as a "big native husky, the proper wolf dog, gray-coated and without any visible or temperamental difference from its brother the wild wolf" (6). The wolf dog acts as a foil to the unnamed man; or in other words, the author seems to insert the dog into the story specifically to show readers what the man is lacking. As readers, we might be impressed by the fact that it's minus seventy-five, but the dog does "not know anything about thermometers" (6). Rather, the "brute [listens to] its instinct" (6), which tells it not to travel on such a cold day. The man, on the other hand, doesn't seem to have any instincts of the kind.
Later on, after the dog has gotten its legs wet, the narrator redraws the line that separates animal and man. The dog recognizes that if it leaves the ice on its feet, walking will become uncomfortable. The dog doesn't "know" this the way a human would know something. Rather, we are told again that this is "a matter of instinct" and that the dog "merely obey[s] the mysterious prompting that [arises] from the deep crypts of its being" (6). Deep crypts of its being? London always has a great way of making primitive instincts sound mysterious and spooky.
The man also knows that he should get the ice off the dog's feet, not from instinct, but from what the narrator calls "judgment," which is quite different. And since the man ends up dying, it's not too much of a stretch to say that there's something lacking in his judgment, compared to the wolf dog's instinct.
In the end, the wolf dog doesn't realize right away that the man has died. The pooch isn't used to seeing a guy sit in the snow for so long without a fire. But when it catches the stink of death coming off the man, it howls for a minute and then continues trotting toward the mining camp, where it knows it can find more fire and food. It sees humans as food and fire providers, not as buddies. He can use them for his survival, but he's not about to sit around and mourn their loss.
The ending of the story is very unsentimental, and it is mainly through the dog's final desertion of the man that Jack London is able to convey that nature is a tough, uncaring thing. It's not exactly how an episode of Lassie would end, but it gets the job done.