Learn Dutch: Wij luisteren naar de muziek. - We are listening to the music.

Because the subject is wij (we), which is plural. In Dutch, plural subjects (wij, jullie, zij) use the plural form of the verb. For regular verbs, this plural form looks exactly the same as the infinitive (the basic dictionary form).

The translation says "are listening", but there is no word for "are" in the Dutch sentence. Why?

Dutch doesn't normally use an "is doing" or "are doing" form like English does. The single present tense verb luisteren covers both "listen" and "are listening". You understand which one it is from the context.

Is luisteren naar a fixed phrase just like "listen to" in English?

Yes! Just like you cannot simply "listen the music" in English, you cannot just luisteren de muziek in Dutch. You must use the preposition naar (to). It is highly recommended to learn these pairs together as luisteren naar.

Why is it de muziek and not het muziek?

Muziek happens to be a de word. There is no strict rule that will tell you this just by looking at the word, so it is best to memorize the article de right alongside the noun when learning it.