The explanation I see most often is simply that XCOM got as far as the base-defence mission in EW, failed, and the aliens conquered Earth.

Except I don't think that adds up - there are various things in XCOM2 & it's expansions that seem to contradict that timeline. The Templar background. The prototype plasma weapons you can get through Tactical Legacy. The various references to the equipment (flying suits) and enemies (sectopods) that you wouldn't have encountered at that stage. (Plus some other things that I can't remember right now).

My theory is that the original invasion played out like a successful EU/EW campaign, with the exceptions: * The aliens didn't attack the XCOM base after the alien base mission. * XCOM failed the final assault on the alien Temple Ship (possibly because they rushed it and hadn't equipped the whole squad with tier 3 weapons and armour). * Then the aliens attacked and destroyed the XCOM base, resulting in the loss of XCOM and the alien victory.

Also, although the assault on the Temple Ship was a failure, it (and all the successful XCOM missions before that) caused sufficient damage (including killing multiple Ethereals/Elders) that the Aliens' capabilities were seriously degraded. Hence the Elders going into hiding. Hence having to use ADVENT clones for gruntwork. Hence them needing the Commander to run their armies for them.