Jacob Weisberg, who has a thriving practice going as an aggregator of stupid remarks that come from political leaders, writes about why Sarah Palin says the inane things she says. The way I see it, he's way overthinking the question. I think it's a matter of just playing to her audience, saying what the people who flock to hear her like.
Years ago, a female stand-up I knew had a routine in her act that I thought was sexist (from a male POV) and stupid and while audiences howled at it, it was the kind of laughter a guy could get by coming out and dropping his pants — a cheap, easy laugh. It was so out of tone and character with the rest of her material (and her as a person) that I asked her one night why she did it. The answer was, "Because people laugh" and that was literally all there was to it. She did it because it seemed to be advancing her career and no, she didn't pause to ponder if she agreed with that message or the philosophy it expressed. It just worked and that, by God, was that. I think of her when I hear Sarah Palin.