It's hard not to look at recent news of Trump's bragging about his very stable genius, and then flash back through the history of anti-intellectualism in this country's political history. The thing I find most common isn't so much anti-intellectualism, not so much being against higher thought per se, but something slightly different. It's resentment at the idea that someone might be smarter than you are, that they don't have to brag about it because they can just go out and be smart, that they are smart enough to lecture you about why you're wrong about something when you know you're right about it, and so the response to that is not to lift yourself up, but to drag the others down and usurp their place. You're the smart one, not them. Or at the very least, you're smart too, and anyone who says you're not is a lying liar.
Trump desperately wants to be thought of as smart — why else would he brag about being that way? — without actually having to be smart. But as someone else once put it, he's a dumb person's idea of what a smart person is. I've known truly smart people who were also egomaniacal about it, but from what I've seen they actually tend to be the exception, not the rule. Most of the smart ones just go out and embody their smarts and put them to use, not hoist them into a trophy case and brag about how shiny they are.
Some people out there are smarter than I will ever have any hope of being, and I don't deal with it by getting annoyed at them. But if you've been trained to see life as a zero-sum game, I guess that's the natural reaction. Of course you're going to want as much of whatever anyone else finds valuable for yourself, even if you have no idea what to really do with it.