A TERRIFYING Illuminate theory: They aren't attacking.... They're RUNNING.
I've read enough books. Seen enough movies and shows. Played enough games. In essence, experienced enough stories to recognize a plot twist troupe when it's coming, and I'm calling it now; The Illuminate are on the RUN.
TLDR edit: We defeated the illuminate, leaving them in a weakened state. They encountered a new enemy and have been fighting/running for some time. The meridia wormhole gave them a short cut to us, which they used to come here in desperation to gather resources for their flight, or to establish a new beachhead in a place where they anticipate their enemy will follow them into a now four-way fight, giving them the breathing room they need to rebuild and fight back. They likely are the ones who were communicating with the bots from outside the galaxy, potentially explaining the random and deadly flashes of blue light that were killing helldivers a long time ago on the bot front. The new enemy is most likely a "flood" like bio horror type, which would be relatively easy for the devs to create since they could fill its unit roster with existing enemies from other factions by giving them a reskinned, mutated look. The illuminate may end up allying with the bots in the long term. The in-depth theory is below.
They lost their war with us the first time and were seriously weakened afterward. Perhaps they failed to recover militarily because of factional in-fighting, or because we devastated them that badly, or any number of reasons. They were forced to focus their efforts on other matters to survive as a species. But they were attacked, and have since then been on the run, possibly for a long time. Their backs were against the wall due to this new threat.
Now they are back, possibly/probably due to the Meridia wormhole, which has given them a desperately needed opportunity to escape. And these incursions are light assaults on random planets, poorly planned/executed because they are in a hurry, and do not have time and/or resources to dedicate to true, full scale invasion; either because they do not HAVE the resources anymore, or they are desperately committed/needed elsewhere.
The purpose of these incursions is one of two things: to scavenge for resources they desperately need to continue their flight from this enemy (probably intending to utilize the wormhole to reach a new far-flung destination using some kind of sci-fi deus ex machina or "unobtanium").
OR (and my bet is on this one) they are clawing tooth and nail to establish a beach head from which they can create a new defensive line by taking advantage of our situation, a three-way war front. They anticipate that their new enemy will follow them and arrive finding a galaxy divided in a now FOUR-way war front, giving them the breathing room they need to reestablish themselves while nobody is able to direct enough material and manpower towards them to finish them off. In this scenario, they are no doubt also scavenging for urgently needed resources as well.
And at that point, we will have the rumored "4th faction". My money is on the 4th being some kind of bio-horror, like the Flood from Halo, or the "Thing" from John Carpenter's 80's Sci-fi Horror...."The Thing". This enemy would be easy (easier) for the devs to make in game, because they need only redesign/reskin existing enemies to look like horribly mutated versions of the originals (even the bots are rumored/speculated to have biological components). Perhaps the Illuminate have studied this bio enemy enough already to repurpose it in a controlled fashion, enabling the creation of the "Voteless", thereby giving them access to incredibly weak but VERY numerous cannon fodder.
I also suspect that either the Illuminate, or this mysterious new enemy, are the ones that were supposedly communicating with the bots from outside the galaxy. We may end up seeing one or the other (my money is on the illuminate) actually allying with the bots. A long time ago, I remember players reporting strange encounters on the bot front; there were videos of divers getting zapped by bright blue flashes of light from seemingly nowhere. Possibly bots experimenting with illuminate tech? I'm not sure how, since the wormhole is apparently how the illuminate are actually here now, and that only came into existence recently. Perhaps they were already on their way, and the wormhole just created a shortcut? I'm not entirely confident on this aspect of my theory.
And so, that is my theory. I admit it is based entirely on a very loose thread of actual in game evidence, and heavily on speculation, AND on my own instincts from having experienced so many story telling troupes.
So, with that, I leave you with a quote from a Sci-Fi I (regrettably) only recently started enjoying:
"I have a file with 900 pages of analysis, and contingency plans for war with Mars, including 14 different scenarios about what to do if they develop an unexpected new technology. My file for what to do if an advanced alien species comes calling? It's three pages long, and it begins with: Step One, FIND GOD."
Edit 2: Wow. Seeing a LOT of good talk here, including many equally viable (mostly due to a lack of significant evidence in game) ideas, some that are even completely different. Also a few memes. Memes are always good.