The ship is stuck in sheets of ice in the ocean when the crew sees a giant figure in the distance going across the ice on a "sledge." (We're thinking "dog sled.")
The next day, the ship's crew finds another man on yet another sledge. Unfortunately for said man (and his dog team), all but one of his dogs are dead. This man also looks like he has one foot and possibly half a leg in the grave.
So the crew brings the new guy on board the ship, rubs his body with brandy, and gets him drunk to warm him up.
(Don't try this at home. This was back before they knew about alcohol, and how it actually lowers your body temperature.)
Walton wants the new guy all to himself to be the friend he's dreamed of having, which is weirdly possessive.
At the end of this letter, he tells his sister that the man is going to tell his story the next day.