Attacker wants to be able to send an especially crafted public key to their target's server's sshd. That crafted key is totally bogus input, a normal sshd would just probably reject it as invalid. The bits embedded into the key are actually malicious code, encrypted/signed with the attacker's secret key.

In order to achieve their objective, they engineered a backdoor into sshd that hooks into the authentication functions which handle those keys. Whenever someone sends a key, it tries to decrypt it with the attacker's keys. If it fails, proceed as usual, it's not a payload. If it successfully decrypts, it's time for the sleeper agent to wake up and pipe that payload into a brand new process running as root.