How does the author make the illusion of Farquahar's survival seem more real than the actual hanging?
Bierce is able to do this through his veiling of the shift from reality to daydream/memory. Bierce blurs this for the reader in order for the reader to believe that perhaps he did actually escape. Bierce puts the reader into Farquhar's mind, and because he is going to be hanged, his mind is full of memories and a desire and wish to escape. The reader, however, wants him to escape so badly that he/she does not remember that it is just that...a daydream, wishful thinking, reflection, memory, etc.
It is also through Bierce's magnificent description that the reader is able to believe that Farquhar has escaped. This dream-sequence is written with specific, vivid details. For example, during this dream-sequence, Farquhar must swim to safety and he hears the bullets whiz by and hit the water. The attention Bierce devotes to describing this scene is amazing and memorable!