Can you help me understand the math of Bell inequality?

I tried to look through the theory and have read/watched several explanations on why local realism does not work, but none of them actually explained the how of Bell inequality - why the local hidden variable theory probabilities differ from quantum mechanics. I also tried to read through the theorem but I just don't get it on my own so that's why I am asking here.

So the question is - how exactly does local hidden variable result in different disagreement rate than quantum mechanics? How does the Bell theorem prove that there cannot be an unknown variable that would result in the same disagreement rate?