Peyton Farquhar does not pay attention to the details which might save his life...had he been observant, he may have noticed something about the uniform of the soldier he welcomed into his home or that he doubled back and headed to the Union army instead of the Rebel army.
Farquhar's sole goal and interest is in being a hero for the war. He jumps at the opportunity to be this hero at the suggestion of the disguised Union soldier that burning the bridge would be a great deed.
Farquhar, consequently, is captured and sentenced to death. He is reduced to only a person who fell into the trap of the Union--not a hero, not a wealthy landowner, not a famous soldier.
His anonymity also parallels the anonymous soldier who visits. Farquhar just takes him at face value...no name, just a uniform of the Rebel army and a symbol of his home in the South. It is part of the brutality of war and the injustice of innocence paying for the mistakes of others.
A central theme of Naturalism is the indifference of Nature to what happens to the individual. This indifference is conveyed by Bierce's objective narration in part I in which only an observation of the setting and character is provided. The focus of the reader falls to the setting and action of the instruments of war who observe military etiquette in "silence and fixity," at least until the man ponders his death as he becomes conscious of his watch ticking. Thus, the anonymity of the character conveys the coldness of war and indifference of Nature.
With this focus now on an individual, the observations of part II are from within the character of Peyton Farquhar as well as outside him; thus, the narration switches to omniscient narration in part II; the narrator knows everything, the inner thoughts of his character as well as his actions. In part III the narrator switches to limited third person as the narrator focuses solely on the thoughts and feelings of one character. With this narration, the reader begins to identify with the character.
However, the stark ending returns the reader to the very indifference of life and the brutality of war. As he is in the beginning of the story, Peyton Farquhar becomes just a man, a prisoner now dead.