I'll try explain it, because it's not that complicated. There are only three moving parts in the classic formula.
You have three doors, A, B, and C, behind one of which has a prize.
You choose A. The host eliminates C and gives you the option to stay with A or choose B. The winning move (50% chance) is to choose B.
The logic is no more complicated. When you choose A, you have a 1 in 3 chance. The host eliminates C and provides you new information. This changes the probability of C from 1 in 3 to zero. Because this is new information, the probability of B changes. The probability of A has not changed. But because the probability of C has changed, and A has not changed because it cannot, because you've already chosen it, the probability of B changes to accommodate. B is now a 1-in-2 chance of being correct because an incorrect option was eliminated. Your choice of A is still a 1-in-3 chance because C was eliminated as a consequence of your selection of A. Given that B is now 1-in-2, B is the correct play.
I get that it's kind of counterintuitive to think about, but the more you know about the probability underlying it all, the more it checks out.
In this context, Balldo knew that a judge gave a bad outcome, and rolled the bones on a better. The parallel is a little strained, but I imagine someone with a better legal grasp could explain it using Bayesian principles.
