From my understanding, in order to retrieve tokens on a side-chain, you would have to send your XMR to some address that would then be unlocked when you burn the side-chain tokens.

How could Monero verify that anything has actually occurred on the side-chain and unlock the XMR?

dEBRUYNE answered the following in relation to a side-chain question:

To prevent issues with the mainchain we would have to implement a cryptographic proof, i.e., the amount of XMR that went into the sidechain would have to be equal to the amount of XMR that goes back into the "mainchain".

I guess I'd like a little more explanation of what the cryptographic proof would be / how it would be implemented?