One example of literary realism occurs when the story opens with the narrator's description of the setting. We get very straightforward images of the "swift water" some twenty feet below the railroad bridge, the placement of the hanging rope, the boards that have been "laid upon the ties supporting the rails of the railway [that] supplied a footing for [Farquhar] and his executioners," the officers and sentinels, and so forth. There are no artistic flourishes or figurative language; instead, these descriptions are, well, realistic, and lack any embellishments that would make the objects and people describe seem anything other than relatively ordinary.
Part of what makes literary realism interesting, however, is its portrayal of regular people whose lives are still filled with drama and interest. As Farquhar is about to be hanged, he opens his eyes and looks down at the water, thinking, "'If I could free my hands . . . I might throw off the noose and spring into the stream. By diving I could evade bullets and, swimming vigorously, reach the bank, take to the woods and get away home.'" These thoughts, the narrator says, "flashed" through Farquhar's brain, indicating the rapidity of his thought in this moment. His thoughts in this moment help to prepare us for Part III of the story, where Farquhar appears to escape in almost the exact way he imagines.
There is nothing overtly supernatural or very strange at work in the fantasy he creates as his body falls from the bridge. Farquhar seems to see things in a less-than-realistic way—in extreme detail, presented in a dreamy kind of manner—but that's because these observations are taking place within his imagination and not in the real world. People who think they are about to die often feel that their lives "flash before their eyes," and this is almost what happens to him, except that his imagined escape is what he sees. This portrayal of Farquhar's imagination running away with him, moving so fast (probably as a result of his adrenaline), is dramatic, to be sure, but somewhat realistic nonetheless.