My guess is that they regarded him as authoritative. The perception that he's the power behind the throne has advantages. What I heard, though, was a man who spent most of the time using the first-person plural and very little letting "George W. Bush" pass from his lips. In other words, the appearance of authority may have come at a fatal price. Although Edwards clearly deferred to John Kerry, he managed both to counter Cheney's attacks and to use Kerry's name in positive statements. Again and again, you heard Edwards saying, "John Kerry will do Y" or John Kerry will do Y." And that's precisely the model of what I think a Vice-Presidential candidate should do in debating his counterpart.
Neither man did a particularly good job of responding directly to questions, but that's to be expected in this era of soundbites that have been rinsed and spun to the point of flaccidity. Edwards managed to avoid ad hominem attacks on Bush, while implying that Cheney is a self-interested liar. Cheney went after Kerry's record with dogged single-mindedness. My favorite comeback of the night was when Edwards said something like, "This administration proves that a long resumé is no guarantee of good judgment." Indeed.