Death, as narrator, functions in two important ways in this story. First of all, by opening his story with the blunt "You are going to die," a theme is immediately established. Death is inevitable. This is a war book. We should be aware that death is inevitable - and expect many (if not all) of the main characters to die. But because Death is telling the story, and the fateful news is delivered from the outset, somehow the preparation manages to set a somber but not sorrowful tone to the entire story.
This tone is also set by Death's second function as narrator of this story. His is an ongoing tale - as Death (personified) is timeless. He tells this story, and at the end of it, takes Leisel's story and carries it with him. Despite the fact that he is a taker of mortal lives, he essentially makes a piece of the humans he respectfully admires, immortal.