What is Farquhar's motivation? Why did he do what he did?

Because he was an "original secessionist," which means he sided with the South at the beginning of the conflict between North and South during the Civil War, he is also described as "ardently devoted to the Southern cause," meaning he is passionate about all things Southern.  However, the story vaguely says that he cannot serve in the war as a soldier ("Circumstances of an imperious nature, which it is unnecessary to relate here, had prevented him from taking service with the gallant army..."). But, Farquhar "did what he could. No service was too humble for him."  Thus, when a man dressed as a Confederate soldier tells Farquhar about the advance of the Union army to the Owl Creek Bridge, and how there is plenty of kindling under the bridge that could be lit and burn down the bridge, Farquhar jumps at the chance to help the Southern cause in any way he can.