In the case of the GUSTAR construction, an indirect object pronoun is required: me, te, le(s), os. Many learners are taught that "me gusta" means "to like" because that is the most common English equivalent. "Me gusta" more directly translates to "It is pleasing to me" in which case the indirect object pronoun is obvious because of the use of the preposition "to". Just as we use "to" in English to express indirectness (e.g. He gave the book to me), it is also used in Spanish when emphasizing the person. For example, A mí, me gusta el chocolate.
In your example, A vuestros hijos les gusta leer, the "a" indicates that vuestros hijos is the recipient of the action, or the indirect object. We cannot leave out the "a" because that would indicate that vuestros hijos was the subject of the sentence, which is not the case. In your example, leer is the subject because "reading" is doing the pleasing, is pleasing to your children.
To explain further with the chocolate example:
I could say El chocolate me gusta. Since there is no "a" (Al chocolate) I am indicating that "el chocolate" is the subject of the sentence and "me" is the indirect object pronoun (Chocolate is pleasing to me.)
But saying A mí me gusta el chocolate, is further emphasizing that I am the one that likes chocolate, not you, or them. Chocolate is pleasing to me.
There are many other verbs that behave in this same way, that requires the same use of "a". Some of these are: encantar, importar, doler, molestar