The quotation at the end of Part One of "An Occurrence at Owl Creek Bridge" serves to inform the reader that Peyton Farquar has not yet given up despite his dire predicament. He still has hopes of freeing himself and returning to his family home.
He unclosed his eyes and saw again the water below him. "If I could free my hands," he thought, "I might throw off the noose and spring into the stream. By diving I could evade the bullets and, swimming vigorously, reach the bank, take to the woods and get away home. My home, thank God, is as yet outside their lines; my wife and little ones are still beyond the invader's farthest advance."
It also foreshadows the events that appear to take place in Part III. The reader is led to believe that Farquar's hopes at the end of Part I do occur: The final part opens with Farquar's apparently successful escape from the gallows. The rope breaks, he frees his hands and swims downstream while eluding the enemy bullets. The reader sympathizes with the plight of this family man who is about to die, and the reader pulls for him to escape.
The description is spread throughout the first portion of the story. The first lines of the story open with the physical description, simply recounting his posture. The third paragraph puts him in his social context, letting readers know what they can tell about him by his appearance. The later descriptions move into his mind and heart, and the quoted thought is the clearest example of this. They show his clear mind, and that he is not self-pitying or focused on fear, but rather, ready to act and concentrating on those he loves.