"My mother likes to read books."
Translation:A mi madre le gusta leer libros.
You need the "a" because "mi madre" is the object, rather than the subject, of the sentence. ( "le" is the indirect object pronoun meaning, in this case, "for/to her"). A way of translating the sentence which might help would be "Reading books is pleasant for my mother".
It does agree with the object doing the pleasing, not the person. The "le" is the reflexive (not sure that's the correct term?) referring to the mother, and the "gusta" is just a singular verb referring to the act of reading. If it were referring to "books" then it would be a plural verb.
I still don't get this, if it had been my mother likes to read, I would understand, but my mother likes to read books, I must be stupid but it seems the books are the things doing the pleasing rather than the reading per se. I will just stumble on, have to fall into getting it right sometime!