Indeed. I now saw the protocol overview on that page.
I think this looks a whole lot like OpenID, in the end. In both cases you're supposed to run some web server software to handle the login process for relying parties. The only major difference is that instead of your identity being an URL, it's an email, but that email is converted to an URL by a convention, and that URL is then used to look up the API's relevant details.
Mozilla's additional trick here is that if your email-like identity's provider doesn't support this protocol (the derived URL doesn't contain a valid document), they offer some kind of implementation which can do the authentication nevertheless. I presume *that* does send emails to the address, and you have to read them and maybe follow a link or copypaste some token from that email to prove that you can read messages sent to that email address.