To be clear, the writing sites don't use Xenforo and I am a member and mod, not the admin/owner, or else they probably would be using Xenforo.
The software they use is open source and the owner has a programmer on retainer so I am not sure if the badge system was part of the OS code base or if the programmer built it custom. Likely the latter since adding new badges seems to require his involvement.
Generally, you don't get anything other than prestige out of badges, but they do award badges to winners and entrants in writing competitions they host (I have picked up a few that way myself) and the top three generally get some kind of prize as well ($$, a free premium membership, that sort of thing).