Explain what leads to Farquhar’s capture when attempting to destroy the bridge.

We do not understand why Farquhar is being hanged until we get to Part 2 of the story, which contains most of the exposition. Part 2 is a flashback. Farquhar is at his plantation with his wife when a man wearing a Confederate uniform rides up and asks for a drink of water. This man is a scout for the Union army. It does not seem likely that he has come to Farquhar's plantation for the specific purpose of tricking him into trying to burn down the Owl Creek bridge. But the scout must be on the lookout for possible saboteurs as well as collecting any other kinds of useful information he can, such as Confederate troop emplacements. He tells Farquhar something which might or might not be true.

The soldier reflected. "I was there a month ago," he replied. "I observed that the flood of last winter had lodged a great quantity of driftwood against the wooden pier at this end of the bridge. It is now dry and would burn like tinder."

There may have been "a great quantity of driftwood against the wooden pier at the end of the bridge" a month ago, but it should have been cleared away by now. The soldier may have been at the bridge much more recently than a month ago. When he sees Farquhar's keen interest in this real or fictitious driftwood, the scout must feel pretty sure that the plantation owner plans to try to set fire to the wood and do serious damage to the bridge.

Farquhar has really set himself up for his entrapment by asking:

"Suppose a man--a civilian and student of hanging--should elude the picket post and perhaps get the better of the sentinel," said Farquhar, smiling, "what could he accomplish?"

The author has intentionally caused Farquhar's wife to exit the scene so that her husband and the soldier can converse in private. She has gone to fetch water "with her own white hands;" that is, as a sign of special courtesy to the Confederate soldier rather than sending a black household slave to run the errand. If Farquhar's wife had been present, the two men might not have talked about the bridge and about the Federal advance. Farquhar might not even have told his wife he was leaving that night. She would have been terribly alarmed if she had known he was going to try to burn down a bridge single-handedly with many soldiers guarding it.

After the disguised Confederate scout drinks the water he rides off, and the section ends with the ominous words:

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 know that Farquhar will be riding right into a trap. The scout will warn the officers at the bridge to expect a would-be saboteur. Soldiers will be posted on the bridge and around it. Some of them will have dark lanterns. All will be under strict orders not to make a sound. Bierce does not have to describe how Farquhar is captured because we can imagine the scene all too vividly. Farquhar hitches his horse to a tree at some distance from the bridge and sneaks through the darkness carrying the incendiary fuel that will provide positive proof of his guilty intentions. Next morning he will be standing on the bridge waiting to be hanged, as described in the opening sentences of Part 1.