It cuts out the necessity for a password manager browser extension to handle stuff like autofill, password generation, etc. Those extensions have had fairly significant vulnerabilities in the past. So you're reducing the attack surface, as well as getting a cryptographic guarantee against phishing (the signature the client returns include the domain that sent the challenge).

Edit: The other great part is that the server just stores your public key, so it's idiot proof on their end. It makes a breach effectively useless, since offline cracking is impossible.