Bierce does two things with this particular description and, ironically, the effects are completely contradictory. In the first place, the detail brings the scene to life and adds it an element of authenticity. By making it so detailed, Bierce makes it seem real, which carries his "trick" along until the moment of revelation. If we knew at this point the scene was fake, then the last line of the story would not have the desired shock:
Peyton Farquhar was dead; his body, with a broken neck, swung gently from side to side beneath the timbers of the Owl Creek bridge.
The other purpose of the description is the opposite. Bierce is giving the audience some clues to foreshadow this last line. That was, when we get to that last line, the whole "trick" has validity to it. Although Farquhar is shot at from close range, he isn't hit:
The man in the water saw the eye of the man on the bridge gazing into his own through the sights of the rifle. He observed that it was a gray eye and remembered having read that gray eyes were keenest, and that all famous marksmen had them. Nevertheless, this one had missed.
This is almost unbelievable - clearly, he should have been hit by such a good shooter at such close range. Giving this clue makes the ending of the story believable, and prevents readers from dismissing the whole text as ridiculous.