The phrase "let alone" introduces something that is larger or smaller than the main point.
In this case, the person is willing to sacrifice his life. Apparently he is being asked to sacrifice only his legs. He's willing to sacrifice his life, a much bigger sacrifice than his legs.
You can turn this around too. He's told he has to sacrifice his legs and that it might kill him to do so. He will not make this sacrifice. "Hell no! I'm not willing to sacrifice my legs, let alone my life!"
Let's say you won a prize and it's a family vacation. You would have been happy with 2nd place, airline tickets, but you won the grand prize, an entire vacation. You would been happy with just the tickets, let alone the entire vacation.
You don't like your husband's cousin very much, but you think his children are even worse. You say "It will be bad enough having him here, let alone those awful brats as well!"