I believe that they're singing about disconnected behavior. Talking without speaking means that they make noise without communicating, and hearing without listening means that they received the sound but did not care what was said. It's a kind of alienation, a sadness. You've probably experienced that, maybe when someone wants to tell you about something on the street, a kind of selling, and you don't want to stay and hear, but he keeps talking. He is not speaking to you, he is just talking, it could be anybody in front of him. And when you say what you think, he may say "yes, yes" and repeat his argument - he's hearing you, but not listening to you. This is an important part of intimacy, closeness between people who care about each other, and I think the song is expressing sadness at the loss of that closeness in society as well as between people who know each other.

Sound of Silence is an oxymoron