The ticking of Peyton Farquhar's watch in section I is significant because of its imagery, its symbolism, and its use as a device to trick the reader into a willing suspension of disbelief during the remainder of the story.
First, the idea that the watch could sound "like the stroke of a blacksmith's hammer" and that it could hurt Farquhar's ears is strong auditory and tactile imagery that involves readers with the story. They imagine what it is like to be waiting to die; they feel as if they are on the bridge with Farquhar as time draws out so that the duration between each second lengthens. It is a brilliant descriptive passage.
Second, the watch symbolizes time. Time is running out for Farquhar--his days, even his seconds, are numbered. Every second becomes precious with death in view.
The most important role that this descriptive passage plays, however, is in setting up the escape and surprise ending in section III. Section III occurs during the hanging. For a brief moment Farquhar imagines his escape, but Bierce wants readers to believe the escape is real. Since readers have already been prepared in section I for the heightened senses Farquhar is experiencing, they are more likely to go along with the description of Farquhar's escape. Bierce lets readers see, hear, and feel what Farquhar sees, hears, and feels when his senses are "preternaturally keen and alert." While swimming away from the bridge, he sees the eye of the sniper who shoots at him. He sees the veins on the leaves of the trees on the banks of the river and the insects crawling on them. The insects on the surface of the water make "audible music." These are obviously hyperbolic descriptions, and readers would regard them with much more suspicion if they had not been primed to accept them by the ticking watch in section I. Thus the powerful effect of the surprise ending depends to a large extent on the way Bierce describes Farquhar's perception of the watch at the beginning of the story.