Everybody knows the public key associated with that private key, so you can verify that the private key did sign this list of attributes.
An ssh keypair is an actual public/private keypair, but a certificate is just signed and encoded (but not encrypted) formatted data.
If an ssh daemon has knowledge of a public key used to sign a cert, and has been instructed to trust that cert, and all the dates are good, then the ssh daemon can accept that cert as proof of identity and allow a login.