Why was Farquhar hanged?

In Part II of "An Occurrence at Owl Creek Bridge" the Union scout posing as a Confederate soldier tells Peyton Farquhar:

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.

"Summarily" means without delay, without the customary formalities such as a trial. Peyton is caught red-handed trying to set fire to the Owl Creek Bridge. His capture is not described in the text, but the reader can imagine the scene. One of the reasons the reader can imagine the capture scene so vividly is that Ambrose Bierce has already described it in considerable detail in Part I.

When Farquhar is talking to the Union scout he asks him:

"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?" 

This shows that he has a strong intention to set fire to the accumulated dry driftwood under the bridge. He might even be thinking about killing the sentinel. Then at the very end of Part II, the reader comes to the ominous lines:

The lady had now brought the water, which the soldier drank. He thanked her ceremoniously, bowed to her husband and rode away. An hour later, after nightfall, he repassed the plantation, going northward in the direction from which he had come. He was a Federal scout.

The reader can visualize Farquhar leaving his horse hitched to a tree and sneaking up to the bridge with a big can of kerosene and some kitchen matches. There are soldiers waiting for him in the dark because their officers have been warned to expect him. Suddenly the scene lights up as the soldiers uncover their dark lanterns. Peyton Farquhar is carrying all the evidence his captors need to convict him of arson. He will be "summarily hanged" the next morning. This is how Part I of the story opens.

A man stood upon a railroad bridge in northern Alabama, looking down into the swift water twenty feet below. The man's hands were behind his back, the wrists bound with a cord. A rope closely encircled his neck. 

When Farquhar walks into the trap, the reader can not only imagine the silent scene with the Union soldiers waiting in the dark and posted in various places near the bridge to forestall any escape, but the reader can even imagine Farquhar's feelings when he realizes he has lost his life, his family, his home, his plantation--everything.

Then Ambrose Bierce, the notorious cynic, plays a sadistic trick on the reader. For a long while it looks as if Farquhar is going to make a miraculous escape. The contrast between most of Part III and the hanging scene in Part I is exhilarating. The reader has been standing in Farquhar's boots waiting to fall to his death, and suddenly it seems as if this is not a story about a man being hanged but about a man escaping that terrible fate. But there is no escape after all. This is realism, not romanticism.

Peyton Farquhar was dead; his body, with a broken neck, swung gently from side to side beneath the timbers of the Owl Creek bridge.