Craig makes the argument that morality must be objective, not in the usual sense of the term, but rather in the sense of being universally binding upon all persons on account of what he calls a "Competent Authority" by which he means the God of Abraham, Moses, and Jesus. Here is the argument in deductive form:
1. Objective morality requires moral rules laid down by God.
2. Objective morality, in this sense, really does exist.
3. Therefore, God exists.
The argument is deductively valid, but both of the premises are evidently false. Craig's argument for the first premise is essentially that morality can only be understood as a set of rules laid down by an authority figure. He begs the question really hard here, but he does it with the flair of a showman and the conviction of a true believer. Craig's argument for the second premise relies on the audience not noticing when Craig makes the subtle shift from the almost universal moral outrage at the examples he provides to the idea the we cannot be properly outraged unless God is as well. Okay, well it doesn't sound at all subtle when I put it that way, but I promise he does is smoothly.
Harris, for his part, tries to make the case that we should not think of morality as binding rules handed down from above, but rather as a set of ideas derived from our best scientific understanding of how to bring about the flourishing (and avoid the suffering) of conscious and sentient creatures such as ourselves. He makes a strong analogy with the field of medicine and the idea of health versus illness. We assume that health is better for everyone, then we use science to derive ideas about how to get there, e.g. stop smoking, do your cardio, eat your vegetables, wear your rubbers, etc.
If you want a better sense of Harris' opening statement and basic arguments, you can have a look at this video or others like it, in which he stakes out his position and unpacks a sort of simplified utilitarianism for the 21st century.
During the rebuttals, I noticed that Craig retreated a bit further into philosopher mode, in which he seems to assume that everyone in the audience is taking an undergraduate degree in philosophy and can understand what he is saying even when he doesn't bother define his terms. Meanwhile, Harris stuck with plain language, powerful analogies, and memorable one liners. He also takes a direct shot or two at Catholicism at Notre Dame. He falls short just a bit, though, when he failed to make it perfectly clear that this debate ultimately consists of a sematical struggle over what it means to act morally. The entire debate can be summed up thusly:
WLC: Morality consists in following rules issued from above
SH: No, morality consists in helping people because we happen to like people.
WLC: No, no no, it is all about binding rules from a Competent Authority.
SH: There is no such Authority, and have your read those rules? They are God awful.
WLC: OBEY GOD'S RULES!
SH: HELP OTHER PEOPLE!
And so forth. Basically, it comes down to the question of whether we are morally motivated by fear of God or by the love of people, and I have trouble believing that anyone showed up to the debate truly agnostic on this issue, because one has to settle the question of whether any gods exist before you can really get on with the moral arguments. I agree with John Loftus that the best anyone can do against Craig is break even, but I have to give Sam Harris major props for very nearly doing so, especially on a topic like morality, where both our language and our intuitions are strongly biased towards a dualistic and theistic understanding.