The "HSM cluster" serves as a redundant "device" which is in Apple's possession rather than yours, but which you must trust to withstand tampering, even by Apple.

The option to record the keys yourself is also described in the whitepaper:

"If the user decided to accept a cryptographically random security code, instead of specifying their own or using a four-digit value, no escrow record is necessary. Instead, the iCloud Security Code is used to wrap the random key directly. "

As I said, I can't actually find this option in my iOS settings. Maybe you have to disable Keychain first?