Well, it's not exactly what they said. The badge details will display on the new site if they are linked to the badges in the backpack, or if the files are uploaded by the users, because the original issue information is always embedded ("baked") into the image itself.
However, as you probably suspect, all those links related to authentication will fail because the original source no longer exists. So it will be impossible for anyone to properly verify the authenticity of the badge issuer or the details of the issuing. It is rather like having a diploma from a university which no longer exists.
A possible workaround would be to put up a redirect on the old domain so that any links that would have gone to user badges there will be redirected to a web page you make with information about the move to the new domain.