Well, he seemed so low that I couldn't say no; then he says with a sort of moan: "It's the cursèd cold, and it's got right hold till I'm chilled clean through to the bone.
Our speaker is a nice guy, and he can tell Sam is depressed, so he agrees to do what wants.
Sam says that he’s freezing to death. Well, we probably figured that out, but he wants us to know just how awful, miserable, and bone-chilling it really is. Service is hammering us over the head here and pointing out, yet again, that cold is a really big theme here.
Yet 'tain't being dead—it's my awful dread of the icy grave that pains; So I want you to swear that, foul or fair, you'll cremate my last remains."
Turns out, Sam hates the cold so much that it scares him more than the thought of death.
What really "pains" him (that means hurts him or bothers him) is the idea of being buried in an icy grave. To solve that problem, he makes our speaker swear to burn his body when he dies. Sam wants him to do it whether it’s easy or hard, whether the weather is "foul or fair."
Again, notice the balance between the cold ground and the hot fire of cremation. That comparison is everywhere in this poem.