What I think Stewart is advocating for (or maybe this just just what I would advocate for so I'm projecting) is that:
We advocate with honesty and sincerity. Don't misrepresent the truth to suit your own viewpoints and don't pretend to be a ludicrous parody of your own beliefs in order to force the discussion in your direction. Ask for what you really want. If you want an end to partial-birth abortion don't say you think all family planners are murders and should be on death row. If you want better financial regulation don't say you want bankers kicked out of their homes so that victims of fraudulent foreclosure can live in them.
We treat our opponents fairly. Avoid ad hominem attacks when they aren't relevant to the topic at hand (the almost never are).
We seek to maintain a productive dialog. Is your discuss convincing your opponent ? Is it convincing an undecided observer ? Is it rousing those in agreement with you to take productive action ? If not then you're wasting your breath and creating meaningless polarization.
We try to understand an opponent's perspective. Do I think our involvement in two wars over the course of a decade has turned out to be foolhardy and ill-advised--yes. Do I think many of the acts we've committed were probably criminal--I do. Do I think ulterior motives were involved--it's very likely. However, I don't honestly believe that anyone in power went home and told their SO, "Baby, I think we've found a way to make money by killing Ahab." Relying on dark and vile imagery to paint our opponents as bad people is intellectually shallow. Most people are not truly evil and even those who are could be prevented from doing bad things if we do what's needed to solve the elementary systemic flaws in our political institutions.
None of those points require advocating a position any more moderate than the one you honestly hold. It simply requires that you seek it in a way that's honest and productive and is more interested in moving the dialog forward than in "winning".
As a mathematics graduate student in the boonies, I'm going to counter point here on behalf of Andrew Jackson's mobocracy. I don't have time to read the Bush books. I simply do not have time enough to keep up on being literate when it comes to any topics The Daily Show deems important.
In college, we talked about how to "read texts." Experiment with which perspective you take when you read new ideas. Look for things missing, look for hidden discriminations.
However, modern speech writers took such a large shit in my mouth that I can't do that logically. It's impossible. I literally do not have enough processing power to consider everyone's problems. In New York City a few weeks ago I had this carnivale experience where ambulances were running and all the hundreds conversations on the street kept going. This was right down the street, someone was obviously hurt but these people have been convinced that this is not an issue for them; wall street is, Wisconsin is, Conservatives are, their job is.
Everyone has a voice, no one has, or is getting, a heart.