Bierce's story is designed as a puzzle to be assembled by the reader; the three parts are each in different forms of narration: omniscient, objective, and third-person limited. This switch in narration changes the focus of the reader from the setting to the character to the actions of the character. Also, the use of flashback forces the reader to arrange the events in order and figure out what has "occurred."
The narrative starts with the execution of Peyton Farquar. The factual account is interrupted at the last moment with a flashback. This leaves the reader at a cliffhanger moment. We continue to read to find out how the events led up to this critical scene. The fact that Peyton escapes his immediate trouble keeps the reader's suspense even longer.