[N]ow, I say, there came to my ears a low, dull, quick sound, such as a watch makes when enveloped in cotton. (6)
The heartbeat could be thought of as each person's personal internal clock. When it stops, so do we. Note that, just after this, the narrator says he knows the sound is the man's heart beating (when he's still alive). In fact, the narrator makes a big deal about knowing this. Go to the next quote to see why we are making a big deal, too.