Er.... well... she was in the process of starting listening to rock music, right?. Or maybe she started listening to rock already, and then gave it up, and now maybe starts over again or considering the possibility.
The only realistic environment I would use that in is telling about some past events coinciding with the years my mom was only starting to like rock music, and gradually listened to more and more of it.
например. когда моя мама начинала слушать рок мой папа играл в одной группе.
I don't understand what's so weird for you. "To" is a preposition, just like на, к, о(б) etc. "Listen" just requires that preposition as a logical connector, just as "осуди́ть на" does, or "говорит о чём-либо". Whether it's spatial, temporal or logical function, it doesn't matter; it just always needs this preposition ("Listen to what I'm saying!", never "Listen what I'm saying") :)