Learn Spanish (Latin America): La banda salió del escenario anoche.

Salió is the preterite (completed past tense) of the regular -ir verb salir (to leave / to go out). Because the subject la banda counts as a single "it" (third-person singular), we drop the -ir and add the regular past-tense ending -ió.

Why is there a del (from the) here? In English, we just say they "left the stage."

The Spanish verb salir translates closer to "to go out" or "to exit." Because of this, you usually need to include the preposition de (from) to show where someone is exiting from. Since escenario (stage) is masculine, the preposition de and the article el combine to make the mandatory contraction del.

A band has multiple people in it, so why isn't the verb plural (salieron)?

Even though a band consists of many musicians, the noun la banda is grammatically singular because it represents one single group. That is why it takes the singular verb form salió instead of the plural.

Is anoche related to the word noche (night)?

Yes! Anoche is a single vocabulary word that specifically means "last night." Words like anoche are great clues that you need to use the preterite tense, because they point out that an action started and finished at a specific, completed time in the past.