As Peter Farquhar travels -- in his mind -- to return to his family, his body undergoes changes from the long walk and lack of provisions and water. Farquhar's body is, of course, suffering from being hanged; his mind is trying to lessen the pain of his death by providing false hope and a dream-like state in which he can spend his final seconds. The third-to-last paragraph, as Farquhar's mind begins to evaporate (seen in the increasing stream-of-consciousness narration) shows how his body has reacted:
His neck was in pain and lifting his hand to it found it horribly swollen. He knew that it had a circle of black where the rope had bruised it. His eyes felt congested; he could no longer close them. His tongue was swollen with thirst; he relieved its fever by thrusting it forward from between his teeth into the cold air. How softly the turf had carpeted the untraveled avenue -- he could no longer feel the roadway beneath his feet!
(Bierce, "An Occurrence at Owl Creek Bridge," gutenberg.org)
Each of these sensations -- which Farquhar believes are resulting from his long journey -- are classic results of being hung by the neck until dead. The swollen neck and black circle are from the intense pressure of the noose, which also provides pressure to the entire head because the blood can no longer leave. The eyes protrude because of this pressure, as does the tongue, which often "thrusts forward between the teeth." While Farquhar did not die of strangulation, but rather from the sharp shock of a broken neck, the symptoms would still be present as the body dangles. Telling is that Farquhar "can't feel the road," as he is suspended by his neck and so physical sensation would be limited, as would any pressure on the feet. In fact, while these symptoms are normally be associated with strangulation, each of them would could eventually be felt if the mind remained conscious long enough after the neck was broken. Farquhar's traumatized brain is simply associating its remaining physical sensations with a dramatized escape in its last moments.