Not an expert on cryptography so this might not be entire correct.

My understanding is that the wallet generates the private key just like any other wallet. That is the only copy of the key. The words you are given are generate from the private key using some algorithm that is pretty standard between many wallets. The code that generates those words can also turn them back into the private key on another wallet.

Think of the German enigma machine in WW2 but one hell of a lot more complex. So complex that it would take the fastest computers on Earth millennia to break it.