Вы- as a prefix has the meaning of "out, out from" (Выйти из комнаты - come out of the room; Выделить подгруппу - designate a subgroup) and also completion (Выкурить - smoke [like a cigarette] to the end; Вычислить - calculate/compute). Here the verb it is going to is брать, which means to take; with the вы- prefix it basically means chose; there were many books in the library, but you chose (selected this one "out"): было много книг в библиотеке, а ты выбрал эту.
The start of the phrase (at normal speed) just sounds like a mess - is it accurate??
It helps to know exactly what you type, just in case there is a grammar error or something.
Why do we use "ты(nominative)" with выбрал in this sentence, while we used "мне(dative)" with the same verb in the previous sentence?
I don't know what the other sentence was. In this sentence, the translation is "You chose this book". Ты (nominative subject) выбрал (verb conjugated for the subject) эту книгу (accusative object of the verb, indicating that it is what was chosen).
If the other sentence has an indirect object (dative case), then it means that the thing was chosen for that object. For instance, Ты выбрал мне эту книгу - You chose this book (for me).