Learn Russian: Музыкант наслушался музыки, которая играла на сцене. - The musician listened to his heart's content to the music that was playing on the stage.

This combination creates a verb of satiation. It means to do an action until you are completely full or satisfied. Here, наслушался means the musician listened to the music to his heart's content.

Why is музыка in the genitive case (музыки) here? Usually, listening takes the accusative.

Verbs that express satiation using the на-...-ся pattern require their object to be in the genitive case. You can think of it as a partitive genitive, meaning he consumed a 'full amount' of the music.

The translation says 'to his heart's content,' but could наслушался mean he was sick of the music?

Yes! Depending on the broader context and tone of voice, на-...-ся satiation verbs can also have a negative connotation, meaning you did something so much that you are now completely sick and tired of it (like 'he had heard quite enough').

Is the verb наслушался perfective or imperfective?

It is perfective. Adding the на-...-ся structure to denote satiation generally makes the verb perfective, because it describes an action that has reached its logical limit or completion (the point of being full).