I'm not sure if there's ever been any lore-based explanation for GMs in FFXIV, but given the fact they're clearly based on FFXI's version which did have an (out of universe) explanation, I'm liable to regard the Eorzean GMs have similar origins. In FFXI from what I remember the GMs lore-wise were a mysterious order of knights who acted independently of any nation, as a kind of ad hoc police force/militia overseeing adventurers. Given the strength of adventurers and that they can take down gods, it's little wonder such a powerful organization existed outside of national allegiances to keep these often unruly and free-spirited, yet heavily armed youths in line.

That smoky effect the GMs have surrounding them especially I believe is a kind of special buff only available to GMs that renders them effectively un-agroable by monsters (hence allowing them to appear pretty much anywhere without risk of being KOd), as well as making them invincible.

Also, given also the fact that the GM run player jail in FFXIV, Mordion Gaol, shares it's name with the same corrective facility in FFXI, the lore surrounding the Vana'dielian GMs is probably also correct for their Eorzean counterparts.