The quote you have indicated comes in the third section of the story, which is when, following the initial introduction of the first section and the flashback of the second section, Farquhar is finally pushed off the bridge and left to hang. Note the quote that describes the sensation that he experiences as he falls and then feels the pressure on his throat:
Then all at once, with terrible suddenness, the light about him shot upward with the noise of a loud plash; a frightful roaring was in his hears, and all was cold and dark. The power of thought was restored; he knew that the rope had broken and he had falled into the stream.
The sensation thus relates to the apparent breaking of the rope and the way that he falls into the water below. However, as the rest of the story makes clear, this is only an escape of the imagination in the few seconds that it takes Farquhar to really die. Bierce seems to be focusing on the psychological impact of death on us and how, even in such a short time, we can create for ourselves elaborate fantasies of escape and safety.