An allusion is when a writer makes an indirect reference to another text, person, or event, in order to enrich the meaning of h/er own text. The writer expects that h/er audience will have enough familiarity with the text, person, or event to which s/he alludes that its connection to the text's meaning will be clear. When Ambrose Bierce references the Second Battle of Corinth, he is relying on his reader to have some knowledge of that Civil War battle in order to place, in better context, the actions of Peyton Farquhar. It also serves as some foreshadowing of the way events in the story turn out for Farquhar. 

Apparently, although Farquhar would have liked to serve as a soldier in the war, he was prevented for some reason. He would have been part of the group that fought in the Second Battle of Corinth. Here, Confederate troops attempted to disrupt Union communication lines by damaging the railroad junction at Corinth, Mississippi, but they were eventually forced to retreat after brutal fighting and only escaped destruction because Union forces did not pursue them. It was a tough blow to Confederate pride. This allusion helps us to understand Farquhar's motivation for attempting to burn the Owl Creek Bridge and disrupt Union communication by destroying the railroad there. It also gives us a hint that it will not work out any better for Farquhar than it did for Confederate troops in Corinth. In fact, it will turn out worse.

The story also alludes to the proverb "All's fair in love and war." The idea is that lovers and fighters are not bound by the same rules of honor that anyone else is. It's a bit ironic here, because -- although Farquhar feels that he needn't play fair with Union troops because a war is on -- the Federal scout who tricked him into attempting to destroy the bridge also fails to play fairly. Acceptance of this idea ends up working against Farquhar rather than for him or the Confederacy.