Harris has a bit more to say on this issue in a long note, where he says something that he seems to think is obvious (although he does also cite Joshua Greene and Jonathan Cohen in support).
The first point he makes, in replying to Daniel-Dennett-style compatibilism, is that "people's moral intuitions are driven by deeper metaphysical notions of free will." He adds:
"That is, the free will that people presume for themselves and readily attribute to others (whether or not this freedom is, in Dennett's sense, 'worth wanting') is a freedom that slips the influence of impersonal background causes. The moment you show that such causes are effective - as any detailed account of the neurophysiology of human thought and behavior would - proponents of free will can no longer locate a plausible hook on which to hang their notions of moral responsibility."
Again, Harris is concerned to claim that the folk believe we are somehow able to act in a way that is not controlled by "background causes" such as our own dispositions (and whatever physical make-up we might have on which these are supervenient) and various events at the neuronal level (or even at a quantum level) that we can't consciously control.
This is a very strong concept of free will that he is attributing to "people".
Once again, I agree with him that we don't possess free will in this sense. Again, though, I really wonder how many "people" Out There really believe that we possess free will in a form anything like this sense and how many would, if taken through the idea, report a view that this sort of free will is needed for such things as moral responsibility.
That is not to suggest that the folk have a concept of free will similar to mine or, say, that of Daniel Dennett. It's more that I suspect most people have an idea that is rather vague, inchoate, and possibly not coherent (which is not to concede that the view that Harris is attributing to the folk is truly coherent, either).
Still, to whatever extent ideas such as Harris is describing are indeed widely held, my own view is that this entails that there's a lot of error around. That is, it looks to me most unlikely that we possess free will in any such sense, and pretty much for the same reasons as Harris puts forward. I'm with him on the metaphysical question as to whether such a form of free will exists, but I question whether as many people are committed to it as he thinks.