FWIW, my own intuitive explanation of Bayes' Theorem -- which may be inaccurate and wrong -- usually begins somewhat like this:

Let's say that, one morning, you walk outside your front door, and immediately slip in a puddle of water and twist your ankle. Did CIA agents put the puddle there just to hurt you ? Well, according to the theorem,

a). That's the wrong question to ask; a better question is, "how likely is it that CIA agents made that puddle ?"
b). To answer that question, you need to keep in mind that puddles can happen for all kinds of reasons (rain, sprinklers, etc.), not just due to the machinations of CIA agents.

Of course, no intuitive explanation is a substitute for math...