Introspection reveals thoughts arising, not a self choosing them
Harris's central method is an invitation to close introspective observation: try to catch the moment before a thought occurs, the instant where you supposedly "decide" to think it, and notice that you cannot locate any such moment. Thoughts, intentions, and impulses simply appear in awareness, seemingly from nowhere, and the sense of having deliberately produced them is layered on afterward rather than preceding them.
He uses this as direct evidence against the common intuition that we are the conscious authors of our mental life moment to moment. If you try, right now, to decide what your next thought will be, you'll find you can't actually generate it through will in the way the intuition implies — you can only wait and see what arises, then narrate it as chosen.
This introspective exercise is meant to be more convincing than any abstract argument, because it asks readers to check the claim against their own immediate experience rather than accept it on authority.
Takeaway: try, for sixty seconds, to consciously originate your next thought before it appears — notice that you can only watch it arrive.