Let’s not discount how many smart/educated bigots there are. Many members of the GOP in the US went to an Ivy League school and are smart in their own right. They aren’t all ignorant, many are just bad people.
I suspect that even they know that this is their own projected insecurities and that they don’t have any real justification. Better question: why do we separate players by sex in chess? Why does that matter? Just make one group and call it done.
FIDE has two competitive circuits - the open circuit where men and women (cis or trans) can compete, and the women's circuit. Players can (and commonly do) compete in both circuits. Ultimately the goal of the women's circuit is to boost recruitment of female players and make competitive chess less of a boys' club. Opinions are divided on whether it's the most effective method.
The recent decision affects AMAB people who want to play in the women's circuit, but does not bar them from the open circuit.
It's a pretty shit decision as far as I can see, but it's good to make judgments on the facts.
One reason I’ve heard that sports divide into groups by gender is that men are afraid they’ll get beaten by women. It happened in early baseball IIRC. (source: hazy memory. Disclaimer: I am male)
Bingo, people are going to make some argument about how it's because they do it to offer women fair opportunities but the real answer is that they think women are stupid or at the very least that men are smarter, realistically they think both.