If it is correct, the user is permitted access, and the stored seed is replaced with the password typed by the user. The n th password can be generated when required using a program, or a sequence of passwords can be printed out in advance. The keys are related, but designed so that the private key cannot be derived from the public key. Only the person with the private key can decipher the message. If you are able to decipher the text with the public key, this proves that it was generated by the person who has the private key.
Computation is slow.
For extra security, the user's private key is stored in an encrypted form, and only decrypted when needed.