In Part 1 of "An Occurrence at Owl Creek Bridge," Peyton Farquhar is waiting to be hanged. In Part 3 he imagines he is in the water struggling to free his hands and try to save his life. Part 2 is like a flashback. Its main purpose is to explain how Farquhar got into his predicament. There is no description of what he actually tried to do at the bridge or how he got caught. The last sentence of Part 2, as well as the sentence preceding it, make the reader visualize--in a flash of insight and understanding--pretty vividly what happened.
An hour later, after nightfall, he repassed the plantation, going northward in the direction from which he had come. He was a Federal scout.
We can imagine how Farquhar collected some flammables, rode to a spot near the bridge, crept up quietly in the dark, poured some kerosene on the accumulated driftwood, then started a small blaze--and found he had run into a big waiting contingent of the Union Army. Suddenly the scene was lighted, not only by his fire, but by lamps of a dozen soldiers in blue uniforms. There was no escape. Farquhar was surrounded. He had been caught red-handed committing sabotage in front of countless eye-witnesses.
Ambrose Bierce does not have to describe this scene, because we can see it for ourselves, and we can also share Farquhar's feelings when he realizes he has been set up and trapped. He knows what to expect because the federal scout told him in Part 2:
The commandant has issued an order, which is posted everywhere, declaring that any civilian caught interfering with the railroad, its bridges, tunnels, or trains will be summarily hanged.
By "summarily hanged" the commandant means "immediately hanged without trial." They have only waited for daylight to carry out the order.
The significance is that it shows how Peyton Fahrquhar was being set up from the very beginning. At the start of part 2, the reader is told that Peyton is a strong southern sympathizer. He owns a plantation, has slaves, and is fairly wealthy. It also says that Peyton wanted to join up with the southern army, but was prevented by some unstated reason. The text says that he "chafed" at not being allowed to fight and "longed for release of his energies." Peyton wants to fight the Union.
The Union scout (posing as a southern soldier) mentioned enough tempting details to Peyton about the Union army at Owl Creek Bridge that Peyton begins formulating a plan to sneak to it and burn it down. It would have been a good plan had the soldier actually been a southern soldier, but since he was a Union scout and returning to the bridge, the scout likely had everybody on alert to expect Peyton's sabotage. It's a tough last sentence because what it means is that Peyton was doomed from the start.