A lot of it comes from just already being pretty deep on the lore, so I know where a lot of these details already are in a more general sense, I just have to review that stuff for the specifics. I often check websites to get a skeleton of things; the Final Fantasy Fandom wiki and Gamerescape tend to give decent cliff's notes, and then it's a matter of finding their sources. This video was a little harder than the Ascian video in that sense, because the Ascian details are typically very clustered; all the Igeyorhm stuff is around all of the Igeyorhm scenes, for example.
With the Blessing video, though, that's harder. You have the Hydaelyn-centric scenes in ARR and the Antitower, yeah, but important stuff is also raised in stuff like the 5.2 MSQ and Ifrit's summoning, that wouldn't necessarily occur to you. When it comes to other languages I've got some friends who generally remember and bring up stuff like 'Midgardsormr is way less mean in Japanese', so I could generally recall and ask about that.
So a lot of my actual research comes from talking about the stuff with people who know that subject better than me. The Novice Network is a surprisingly good angle for it; I can call out there if I'm struggling to figure out exactly where something was said, and usually get someone who knows it well enough. The Warriors of Darkness scenes are a great example of that one; I knew generally-speaking that it was mentioned Ardbert killed Ascians prior to Shadowbringers, but someone in the Novice Network was able to mention specifically which patch.
What can also be very helpful, weirdly, are people who think they know what they're talking about. Someone trying to prove an especially absurd 'WoL is tempered' variant is actually the one that pointed me to the Ifrit scene, mostly because they misremembered it. A theory doesn't come from complete nowhere, so it's useful to dig through a claim to find what they think is the proof.
The hard part, in all honesty, is the Encyclopedia Eorzea. I'm Australian, so I can't actually get copies of it; only the Japanese version gets shipped here. But I do have an incomplete PDF lying around that lets me check a few things, and a lot of the more out-of-nowhere things you hear tend to come from there. I can often hit some dead-ends because of the fact I can't check it directly, though; I remember hearing at some point that Ramuh is speculated to be an interpretation of Rhalgr, for example, but I can never find the passage that supposedly says that.