A signed message means the key owner approved the message. The frequent use is to sign payment authorization, and this is transparent inside the wallet.
Minor nit: this is confusing signed messages with ECDSA signatures in general.
Bitcoin signed messages prove the recipient of the
address (not the key) agrees to the content of the message signed.
The recipient may not have/control the key behind the address, and may have no authority to speak for a transaction signed by the same key.