This story starts out in the thick of it with a man waiting to be hanged from a bridge. At first, we don't know the details of who he is and what he's done to get into this situation. It takes a few paragraphs to get the gist, with Bierce choosing to ignore the classical narrative model in which the story and the narrative match each other step by step. In other words, it's not your standard beginning→middle→end affair.
This isn't an accident or a case of sloppy writing, either: by starting off with a guy waiting to be hanged, Bierce chooses a scenario that grabs the reader's attention and fires up some interest.
As we read the first paragraph, we find ourselves asking a bunch of questions about who this guy is and how he's gotten into this situation. We soon get our answers, as Bierce follows up the story's dramatic opening by describing the man in question and giving us some background info.
Having learned that he's a married man named Peyton Farquhar, we rewind to the event that ultimately led to him waiting to be hanged. It turns out that the story is set during the American Civil War, and Farquhar is rooting for the South. Having received a tipoff that the Northern troops were repairing the Owl Creek bridge so that trains could travel across it, he had gone to the bridge with the aim of burning it down. Unfortunately for him, however, the guy who gave him the info about the bridge was from the opposing side and was setting him up. Ouch.
Now that we have our background details, we return to Owl Creek Bridge to witness Farquhar being hanged. Miraculously, though, we find that the rope has broken. We consequently follow Farquhar as he flees from his captors and desperately makes his way home.
Throughout this journey, Farquhar feels seriously disoriented and finds that there's something weird about his surroundings—despite being the region in which he and his family live, it's giving off major Twilight Zone vibes. He notices, for instance, that the forest is much bigger and wilder than he'd imagined, and that the road towards his home seems empty and untraveled. Even the stars seem alien. Still, he carries on his journey until he finally nears his destination.