Hi. You are doing much better than me in not being wordy!
I don't like it either, to the point I refuse referring to it as "Broody Jail" or any kind of jail etc. I am about to build one called the "Broody Box" or something a little nicer.
Incubation Behaviour, nicknamed broodiness or gone broody etc., is when a rise in the hormone prolactin occurs in a hen. This can be triggered by range of factors (like eggs piling up in the nest from not being collected), but is definitely associated with a period of warmer weather. Some chickens are more prone to it than others, regardless of whether the breed is bred to "not go broody" or vice versa.
The reason for keeping the hen inside a wire cage raised off the ground, is to allow cool air to pass underneath her. This helps to cool down her belly and subsequently slightly reduce her elevated body temperature, to cause the level of prolactin to subside. It needs to occur for at least two days straight.
A much more harsher and archaic treatment is to dunk to the poor thing into a barrel of water! I do not recommend that at all.
Simply wiping her body with a cool damp towel is not enough. Placing frozen water bottles underneath her will just cause her to sit somewhere else. (But they could be placed underneath the wire cage?)
You see, during broodiness a hen will squish down in the nest or anywhere like a pancake, to ensure whatever eggs underneath her are incubated effectively. But that also maintains her body temperature. It is a different to position to her laying eggs, where her tail is somewhat raised.
Pancake Chooky Clover:
View attachment 4088721
The wire cage allows for any poop to fall away from her, so she is not sitting amongst it, thus ensuring she stays clean. It is also important to provide plenty of water and food so she can get re-nourished. Just place them in the cage with her.
It does feel mean to deny her access to the ground for dust bathing, but what is most important is her health.
If you like, as I do, place a small log inside the cage. It will encourage her to roost, which helps to break the broodiness.