Listen to these pronunciations of Анна then come back and listen to the computer voice say она.
To be honest, there is no historical account of her actually saying this, much like there is no account of george washington's cherry tree incident. Both of these are stories attributed to history to describe the people. She probably had that type of attitude, but the actual phrase likely never happened.
My wife, a native speaker, explained that this sentence generally means to let her read books if she wants to. It could also be given as advice, as in she's struggling in her readings so let her read books. It can also, but this seems to be a pretty fringe case, be a proclamation as in "let there be light", or something like "may she read books" (note that the grammar matches the russian grammar here). However, it could not be used for your first example where you specifically want to command someone to let her read books. In that case you would use дайте ей читать.
Where does Пусть come from? I have a conjugation table for пускать / пустить = "to let, allow, permit; to let go, release; to let in; to launch, start, set off" and Пусть doesn't appear anywhere in the table. The imperative forms are:
ты пуска́й / пусти́
вы пуска́йте / пусти́те
"Пусть" is not a verb, it's a particle. So while it is derived from the verb "пускать", it doesn't follow the conjugation table. As for the exact form it takes, I didn't find any info on that, but my guess is that that's how the imperative used to form in the past. Later the language gradually changed, and the actual imperative became "пусти" while the particle remained the same.
I can accept - I have to accept, that is - the idea that some words are shortened through colloquial use. It's just odd that Duo would introduce an imperative particle before introducing imperatives - and do so without a word of explanation.
Well, maybe it's not so odd, but actually more typical of the way Duo "introduces" new material, i.e., without much by way of explanation. I suppose we can be thankful that Duo doesn't do that very often.
I interpret the exercise as "Let [it be allowed that] she read the books", so она is the subject of the verb читает and thus in nominative case. The thing being order is "she read" (not just "she/her") and that sentence fragment is treated as an ordinary nominative-verb sequence.
In this sentence, the word "пусть" seems to be stressed. Is this normal for such constructions? I think about a situation like this: Anna reads books. B: Anna! Put these books away! C: "Пусть она читает книги."
Or is it something more like this: Anna reads books. B: I do NOT let you read books. C: "ПУСТЬ она читает книги!"