On registration, a keypair is generated, then the private key is encrypted with the long-term key burned into your security key fob or hardware. The encrypted blob is sent to the server and stored there.
On authentication, after you enter your login, the server sends the encrypted blob and your security key tries to decrypt it with the long-term key it has. If it succeeds, it then request a challenge from the servers, signs it along with the server name and timestamp and sends back to the server. Server validates the signature and if it’s good, log you in.
Expanded: As long you as the user has the security key fob, you can login. You should have 2.