In Turkish, there is more than one way to indicate the possessive. If you wanted to start the sentence with "O," as you're asking, then the word "kitap" would have to be in the accusitive (kitabı). However, you can see here that this is not the case.
Here instead, the lesson is introducing you to the grammatical concept of using the locative suffix (de/da/te/ta) to indicate possession. So here, the word "Onda" literally means "at her" and so the sentence literally translates to " There are three books at her," or, more naturally, "She has three books."
j.wilder, I think we have learned three ways to indicate the possession "to have" until now. "She has the book" = " kitap onda" and = she has three books"= onda üç kitap var". And "she has a book"= "onun kitabi (pointless i) var" where "kitabi" has the possessive suffix (the same form as the accusative one in "kitabi okur"= "she reads the book").
"Number + plural noun" is only every acceptable when referring to a "team".
Üç silahşörler (The three musketeers) ; Kırk Haramiler (the Forty Thieves).
Certain district have such names too. For instance, in Ankara: Beşevler (The five houses) ; Kırkkonaklar (Forty Residences).
Other than those instances, you will always be expected to use singular after numerals.