Message signing is the action of signing a cryptographic message using a private key and its associated address, to prove that you have access to the address. Other people can verify this message by checking the signature against the address to see if they correspond to each other.
This can be used to prove that you own the keys to a specific address or to demonstrate possession of a certain amount of BTC or ETH. Upcoming MiCA legislation in the EU will require Crypto-Asset Service Providers (CASPs) to verify whether a self-hosted address is effectively owned or controlled by that client for transactions above 1.000 EUR. So, you might be required to sign a message before you can withdraw funds from a European exchange
Technically, this feature will implement BIP137 for BTC and EIP191 personal sign for Ethereum