that only I can sign messages with, and anybody can prove from looking at the signatures I create that I really do know, without them actually finding out what x is. This is called a zero-knowledge proof, and it might be a bit tricky to understand at first, but if it’s too difficult to intuitively understand then just remember this:
The Web 2.0 backend of the app generates some random message to sign. The example message shown in the most recent image is “Your SnowApe code is 9uc2w. Happy playing! ;)” The Web 2.0 backend now assumes that this party really is the owner of the private key it claims to own. It can then provide authentication credentials to the user in whatever fashion it so desires.
Finally, the user may use bad passwords, or may have difficulty maintaining their own set of passwords. By using cryptographic signatures, the user doesn’t have to send a password anywhere, and instead sends a signature which doesn’t compromise any private information. Additionally, since the custodian doesn’t store any passwords, there is less at stake in the case of a hack. The very biggest benefit, though, is that the user no longer needs to manage a set of passwords.