What pieces of information about Farquhar's background explain why he would risk his life on such a dangerous mission in "An Occurrence at Owl Creek Bridge"?
This quote explains the first part:
"Circumstances of an imperious nature, which is is unnecessary to relate here, had prevented him from taking service with that gallant army...and he chafed under the inglorious restraint, longing for the release of his energies, the larger life of the soldier..." (Bierce).
This sentence from part two basically spells it out for the reader. He wanted to join the army, but for some reason was unable to. He wanted to because everyone else would have been, and there was probably some dishonor in staying home and not joining the cause of the South. The word inglorious supports this; he feels staying at home is unfavorable. He also years for the "larger life of the soldier" (Bierce) meaning he feels that staying at home makes him feel a bit small.
Deeper reading/reasoning also suggests that he is a plantation owner and has something to lose if the South loses the war.