A lot of people seem to assume that she read it in the History of the Time War book, which would go along with her whispering 'So, that's who...'. At the end when she's talking to the Doctor about forgetting, she tells him about the library, but she only starts telling him about finding out his name a few ecods later and all we get to hear her say is 'I have seen it, in one tiny little corner of that..' before the Doctor silences her. We never really learn tiny little what. I might be wrong about this, but I think there's a good chance she didn't see his name in the book (it only said the Doctor in the book), but somewhere else. Or am I lookingtoo much into this?

Maybe there was a picture of The Doctor in the book, who's to say the book is a normal book so maybe it has words and pictures or even like Harry Potter, a moving picture book

But why would the Doctor's real name be written in a book, especially in one that persumably isn't Gallifreyan if it's uch a secret? Unless it's about the hidden in plain sight thing.

You are

sorry but I really don't know

Perhaps there is some other way that Clara knew the information. If so, there has been no hint or clue of what that would be. That she read it in the book is the only reasonable conclusion we can draw for the moment.

As it is, it appears that the book (or whatever) told her that he originally had a name but that he'd changed it, which is what she asked him at the end. It's also probably the reason that she knew who the book was talking about.