The scene begins with Gilly (Hannah Murray) reciting a list of seemingly inconsequential factoids to Samwell (John Bradley) while he tried to study, tidbits recorded by the apparently super diligent High Septon Maynard.
Then she asks him what the word “annulment” means. Sam explains it’s when a man ends a marriage. Gilly says that this Septon issued an annulment to a certain Prince Rhaegar — you know, as in, Daenerys’ brother Rhaegar Targaryen (!) — and then married him to somebody else at the same time (!!). In secret (!!!). In Dorne (!!!!).
Then Samwell totally interrupted Gilly just as she might have been about to blurt out a massively important revelation.
Now, this scene didn’t connect every single dot for us. But it doesn’t take much to assume this means Rhaegar actually married Lyanna Stark — and didn’t just get her pregnant out of wedlock like the show has led us to assume (so the assumption that they actually ran off together and were in love was likely correct). The series has also avoided explicitly saying Rhaegar is Jon Snow’s father and has only confirmed Lynanna is his mother, but that Snow is Rhaegar’s son is a perfectly reasonable assumption to make (especially when the guy is now petting dragons like it’s no big deal at all).
So, Jon Snow would not only be a Targaryen, but also a legitimate heir. In fact, as a male, his Iron Throne claim would actually trump Daenerys’ claim.
And now, Jon, Daenerys, and Gendry all have Iron Throne claims and are all hanging out together (Gendry has one if you consider Robert’s Rebellion legal).
Let’s put this another way: If all the assumptions are correct…what Jon Snow and Daenerys Targaryen have believed all their lives about themselves isn’t true: Jon Snow is actually not a bastard at all, and Daenerys is not the rightful heir to the Iron Throne.