A man is talking to the darkness (put many possible interpretations for darkness here, I'll let you pick one) about a vision he had, and just can't shake. The vision has been "planted in his brain" so it obviously affected him deeply. In this verse you first hear the recurring paradox "the sound of silence", here used to represent his thoughts (i.e. they are given sound by him, but otherwise they are formless).
He then begins to describe his vision (or his "restless dream"); as he walks down a dark, lonely, cold street he suddenly sees the "flash of a neon light". I think this neon light shows that the vision is one of contemporary culture, used to represent commercialism and flashiness (think Las Vegas). Had it been a candle light, the songs meaning would bevery different. The sound of silence here could be the quiet calm of the night (whihc is split by the light) or it could still be his thoughts (which are "touched" by the light).
The light brings him a vision of thousands of people, all caught in paradox; They are "talking without speaking", "hearing without listening" and "writing songs that voices never share". These people are going through the motions of life, but not really experiencing it. They can speak and hear and write songs, but they don't get the full effect of "listening" or "speaking". I bet you know people like this. the kind of people who talk and talk and talk, but never speak anything meaningful. They are shown in the neon light, reflecting that modern society helps these people go on with their senseless lives quite easily. Here the "sound of silence" is the detached rut that these people have put themselves in that no one dare break (by "speaking" or "listening").
The singer tries to reach these people, telling them that this unspeaking, unhearing silence will consume them like a cancer. He wants to reach them, so that he can teach them how to "listen" and "speak".