Their main focus seems to be on the Abyss instead of New Eden, though they’re sure to lay claim to what they deem important here - the Sleeper Caches, the Observatories, the Empress…
On the other hand though, I’ve never seen any of the Drifters rep each other or receive reps from anything - could be related to the very weapon people fear in fights with them 
As-is, I don’t think conquest of the Empires is their goal (at least for now) - the Triglavians seem to be a very major focus for them, and I’d wager the massive Drifter fleet seen in the Hilen Tukoss video was a sortie that descended into the Abyss through the Nexus gates somehow - with the Drifters we’re fighting ourselves down there being those same ships.
As for the “why” of their attack on the Trigs, I’d suppose it has something to do with both of their identities - if the Triglavians are in fact a de-TonyG’d Enheduanni (explaining some stuff here) with the Drifters in fact being the Others, their aggression toward them would match up very well.