Learn French: Elles aiment la musique et elles dansent souvent.

You don't! For regular -er verbs in the "they" form, the -ent ending is completely silent. To your ear, ils aiment sounds just like il aime, and elles dansent sounds exactly like elle danse.

Why is the "s" in the first Elles pronounced like a "z", but the second one is silent?

It's because of a rule called liaison. The first Elles is followed by a word starting with a vowel (aiment). To make the sentence flow smoothly, the normally silent "s" connects to the "a" and makes a "z" sound. The second elles is followed by a consonant (dansent), so no connection happens and the "s" remains silent.

Why is there a la before musique? The English translation just says "They like music".

In French, when you talk about your general likes or dislikes using verbs like aimer (to like) or détester (to hate), you must always use a definite article (le, la, l', or les). You can't just say aiment musique; it must be aiment la musique.

Could I say elles souvent dansent (they often dance) instead of elles dansent souvent?