Poe opens "The Pit and the Pendulum" by describing the narrator's sickening agony of awaiting his sentence. How does this beginning contribute to the meaning of the story?
Obviously, this answer depends on what you believe the meaning of the story to be, so I will answer with my own opinion--however, it is by no means the only one. In my own opinion, this story is meant to demonstrate the fear of the unknown.
While his senses are limited, as he tries to describe the things that were happening to him despite his limitations, he remarks upon the "shuddering terror" that overwhelmed him when his ability to really think returns to him and he realizes that he does not truly know what is going on.
When everything returns to him, he describes his reluctance to open his eyes:
... while I strove to imagine where and what I could be. I longed, yet dared not to employ my vision. I dreaded the first glance at objects around me. It was not that I feared to look upon things horrible, but that I grew aghast lest there should be nothing to see. At length, with a wild desperation at heart, I quickly unclosed my eyes.
He spends the next period of time stumbling about in terror, nearly falling into the pit that he had no way of knowing was there because he could not see a single thing. He remembers terrible rumors he had heard about what goes on in the dungeons and he fears that they are all true and that he is about to suffer a horrible fate (which he is).
Eventually, the lights come on, and he is finally able to see everything around him; and with that, the narrative becomes far less fear-driven and he sounds far more rational. So the main reason he was afraid was because he had no idea what was happening to him.