You need slightly more understanding to go with your reading:
"Authentication" means "verification that you are talking to who you think you are talking to".
In SSH, before you send your authentication information to the server (for it to verify that it is talking to you), the server first sends it's own public key, and specific message signature with the corresponding private key. Your client checks if the public key is already known as belonging to the server (by checking known_hosts), and if not, asks if you are willing to trust the key. If you say yes, the client computes the same specific message, checks if the signature sent matches the message and public key. If this succeeds, then your client has successfully authenticated the server (verified that it is right server), and can trust that it is not a "man-in-the-middle" trying to steal your password.
After this, your client sends your authentication information to the server, and the server looks up your password in the password database, or verifies your public key, or whatever, to check your info. If this succeds, the server has authenticated your client, (verifed that your client is under your control), and can trust your client to run commands under your user id.