It uses asymmetric cryptography. You sign a login request with the locally stored private key and the service verifies the signature with their stored public key. The PIN on your device is used to unlock access to the private key to sign the login request.
Passkeys aren’t a full replacement in my opinion, which is what DHH gets wrong. It’s a secure, user-friendly alternative to password+MFA. If the device doesn’t have a passkey set up you revert to password+MFA.