I think you're on the right track, don't use analogies, explain how it really works in steps. 

There exists a mathematical relationship between a 2 numbers, a private key and a public key where you can prove you have the private key without showing it.  This proof is called a signature.   If they believe that is a real thing go further down the rabbit hole.   The fact that you cannot determine the private key from a public key and the fact that these are massively large numbers are important.   Not sure you need more than that, could go into some details on how transactions are relayed and verified, leave it up to them.

The details are complex but the concepts have to be agreed on for the most part, not understood in detail.  Let's agree my car moves when I push the gas. Ok, now you can dig as far as you want into the details of how that happens, but for our lesson today let's move on to the fact that the wheels turn when I rotate this wheel.