A dignitary is a high-ranking official, often from another country, whose status means he has to be treated with a special measure of respect or dignity. "Death is a dignitary" is a metaphor that likens death to a high-ranking person who has come for a visit.
This refers to and explains the careful rituals of death that the Union soldiers are following as they execute Peyton Farquhar for trying to burn down a bridge. They treat him with respect as they stand at "'parade rest,' the butts of their rifles on the ground, the barrels inclining slightly backward against the right shoulder, the hands crossed upon the stock." Everyone is motionless and silent as it becomes time to witness Farquhar's hanging. Although Farquhar is the enemy, the soldiers understand the dignity of the moment and the commanders show that they take the death seriously.
The narrator offers a detailed description of the scene of the hanging, summing up its solemnity with the metaphor of a dignitary. It sounds like something Farquhar himself might say, as he is depicted in the story as a gracious Southern gentleman who likes to follow the correct forms. Nevertheless, it is the wording of the omniscient narrative voice, a voice that is especially careful to make clear that it is death, not Farquhar, that is being treated as a dignitary.