Because they would have preferred to win
Why do people sometimes cry when they lose a chess game?

because I was like trying to find my blue socks this morning, uuuuhh, and they weren't clean so like I had to pick out the brown ones, uuuhhhh, and like well they didn't match with my blue corduroy jacket and, uuuuhhh, then I lost a chess game, uuuhhh, and Belinda said she didn't like me anymore, uuuhhh, and then like
...to be continued.

People cry after experiencing failure or disappointment in all areas of life-family, relationships, career, or leisure activity. Crying releases stress hormones, thus making you feel better.
because they can't admit that they made a mistake
Personally, I disagree. I usually cry after seeing my own mistakes. They're usually completely ridiculous mistakes as well, and it makes me upset that I made a fool of myself for not seeing the moves I should've made.
I always get upset while playing chess when I feel stupid. It's not that I'm not admitting to making a mistake, it's that my own mistakes make me feel like a failure. I get that it looks pathetic, but we all handle stress or frustration differently. It's nice to let it out I guess.
(Aight that's all, I can't really speak for others, because everyone goes through it differently, but my case personally doesn't match with what you said.)

When I was younger, I used to tear up after losing. But it wasn't from sadness - it was from anger.
I wanted to physically rage and damage things, like how Eric Hansen smashes chairs around after losing. Holding all that fury in made my eyes water from the strain ...
(Thankfully, I've mellowed out with age. )

Emotional immaturity, not being content with their lot, realization of disproportional perception of skill, wack tear ducts, any other plausible reasons?

When I was younger, I used to tear up after losing. But it wasn't from sadness - it was from anger.
I wanted to physically rage and damage things, like how Eric Hansen smashes chairs around after losing. Holding all that fury in made my eyes water from the strain ...
(Thankfully, I've mellowed out with age. )
Stil1: I wanted to physically rage and damage things, like how Eric...
Me: WAIT WHAT?!
Stil1: ...Hansen
Me: Ohhhhh
(yeah why do they cry?)