Because "fratele tău" is the nominative/accusative form. Here, since "your brother" is the recipient of the action "to ask", we are in the dative form. Morever, it needs to be articulated because of the possessive. The articulated dative form of "brother" is "fratelui".
I think that "fratelui tău" is in this case in the genitive form, with the meaning "you ask for a book from/of your brother". Genitive is case about possession (of/from) and dative is the article used when describing to whom/what reffers something (to/for)