Like all questions of authentication, it's a question of trust.
Authentication is essentially someone proving they are who they say they are - we most often do this by providing a shared secret, a password. i.e. when an unknown user connects to your trusted server, they identify themselves and provide the secret. The server trusts the user is who they say they are, because they know the secret.
You see this in the other direction whenever a user connects to a server they don't necessarily trust, such as their bank (they want a higher level of assurance) displaying their bank account #; something a third party wouldn't know. You get this to a lesser degree whenever you log in and see your name, or avatar or whatever up on the screen - it engenders a feeling of trust, though it's much less formalised than "I trust that password is correct".
You get the full formal exchange of trust whenever you log into something that uses multi-factor authentication - if I login with my username and password, the server now trusts me, if I enter the key on my RSA token, or the number I got in an SMS, I now trust the server in return.