Cheque protocol solves a common multi-sig use-cases
An attestation is a signed message about a certain fact. A cheque is a special case of such a message issued to an identifier about some actions to be performed by smart contract (like cashing out some crypto asset).
For example, a cryptographic equivalent of the following messae would be a
person identified by email@example.com can redeem 1 amount of XToken, signed
In TokenScript, a cheque is modeled with a TokenScript just like normal attestations, and deliverred off chain, in some encodings (typically, Magic Link), which the recipient uses to execute the actoin associated with the cheque.
To use cheque on the blockchain it's necessary to cryptographically hide the identifier so that even an adversary who knows the identifier could not link a blockchain transaction to the identifier. Cheque protocol achieves this through Pedersen Commitment and Zero Knowledge Proof.
The cryptographic details are in protocol description.
There are lots of real life use-cases of Cheque Protocol
The actual protocol is here send token by identifier attestation MVP.