The usage of the particle WA is one of the most difficult things Japanese learners encouter. WA is usually explained as a "topic marker". The English translation of WA would be " as for", "speaking of", etc. So the sentence in question, おんがくは、あまり ききません (ONGAKU WA AMARI KIKIMASEN) can be translated as " As for music, I don't listen to it so much".
You are wondering why it is not おんがくを あまり ききません (ONGAKU O AMARI KIKIMASEN). In this case, it means basically the same thing; "I don't listen to music so much", using おんがく (ONGAKU = music) as a direct object. (を (O) is a direct object marker.) It is true that the subject is "I", but you know that the Japanese subjects drop automatically when it is not necessary. That is, if both the speaker and listener know who the subject is, they do not use it, unless they want to emphasize it for certain reasons.
Now, the confusing part is that a topic is often a subject of the sentence as well. There is a subject marker が (GA), but it is often replaced by the topic marker は (WA). If you don't understand why use WA not GA, or why GA not WA, and so on, please submit a question. I can help you with that confusion, too.
When the particle "ha" (as opposed to the feminine speech "wa") is in a sentence that doesn't have a "ga," the topic doubles as the subject. In those sentences, it is often a useful bridging move to replace the "ha" with an appropriate pronoun. "watashi ha baka desu" becomes "myself, I am an idiot." After using this process only a few times, it becomes easy to assimilate how it all works, and "watashi ha baka inai" becomes "I am not an idiot." (and being not a very good student of Japanese, I am guessing that I got that last one wrong somewhere.) For those who have perhaps studied other languages, the "ha" particle can be viewed as marking the genitive case.