I get the idea the game wants me to think that helping Judy with her mission about prostitutes or whatever is the ethical choice but tbh I'm not really clear on what's going on and don't really want to listen to the dialog again.

Can anyone give me the perspective, in a nutshell, why I should my character help Judy roleplay-wise? Why should I care if she succeeds or fails in whatever this endeavor is?

Basically Judy wants the prostitutes out of the hands of the Tyger Claws, who runs Cloud and previously owned Lizzie's Bar.

She wants the Mox (the "prostitute gang") to be in charge of their own lives. There is no noticable change in this aspect later in the game regardless of outcome.

But note that the outcome of Judy's questline when doing the mission at Clouds and onwards decides whether you get to romance her or not (only possible as female V).