In Public-Private Key Cryptography, you can use your Private Key, (without sharing it with anyone), in a way to create a message that is completely tied to your Public key.

Anyone can look at that message, and your Public key, and deduce that the only way that message could be generated is by someone who knows the Private key, even though they don't know what the Private key is.

The fact that the message can be confirmed with your Public key proves that you must know the Private key, and must be the owner of the Public Key and the Wallet Address.

Think of it as a lock that is left closed all the time. You can unlock the lock anytime with your secret key, without showing anyone your key.

Anyone can look at that lock, and come to the conclusion: "This lock must have been opened by the owner who has the key; no one else could have opened it", but you still never shared the key with anyone.