It is generally not, most tournaments anyone can enter. However, there are women-only tournaments because for a lot of social and historical reasons, men dominate the open tournaments.

Interesting side-note fact on the power of social norms: if a woman and man play without knowing each other’s gender the woman will be more likely to win than she would if either party is made aware.

I believe it's done mainly to encourage women to play. More than any genetic differences like other sports.

I feel like as it makes very little difference, so letting trans people play either doesn't really matter.

Because different genders taste cheese differently obviously duh. Don't want to give them NBs an unfair advantage in the Roquefort round

(Serious answer - I think it's to try and combat entrenched sexism in the sport? There aren't many women in chess and by making a space explicitly for them you hopefully create a safe space that can encourage more women to take up the pursuit. As it's a social perspective thing, AGAB therefore really shouldn't matter because the point is to go "look women!" Not "women are inherently better/worse and so we should segregate on gender")

To add to squids answer: There isn't a segregated mens and womens category. There is an open category and a women's only category.

What happened in the open category is that because the societal pressures and social constructs that disincentivized young girls to play, women weren't placing high in the open category. (Because top players end up being top players because they started when thwy were 5) This leads to a feedback loop where young girls see less women in the sport and get reinforced that it is not for them so don't pick it up at a young age, so less persue it and get good, so less women are seen at high levels, etc.

So then comes the women's category to combat women not feeling like they belong in that space. Women can compete in both the open and women's categories.

But because it is an intellectual thing mostly, barring transgender women is ridiculous. In athletic sports you could almost try to argue that a woman that went through male puberty could be stronger(ignoring how estrogen weakens them and they cannot compete in the men's category anymore). You could try to make that argument in athletic sports (and it is a different discussion to this) and almost seem logically consistently on the surface level if you don't think about it any further than your fox news talking points, but what is the argument here? If a woman went through a male puberty they were possibly socialized as male and weren't told as a kid that chess wasn't for them and so they have an intellectual advantage over cis women?

I don't get it. It seems like, just with athletic sports, it is not about the sanctity of the sport or about fairness, it is about banning trans people from public spaces and policing what women can be.

it isn't. the main league and FIDE scoring have no such restrictions. the women's scoring and events were created separately because chess is heavily dominated by men for whatever reason. i think trans women can compete in the main events which are not gender restricted, just as non-trans women are welcome to.