"Music soothes the savage beast" is derived from a William Congreve play, "The Mourning Bride." The actual quote is "Music hath charms to soothe the savage breast," which means essentially that music helps to calm agitated emotions.
"The Mourning Bride" is a five-act tragedy published in 1697 that concerns a secret marriage between the children of two feuding families. Congreve's plays have contributed a number of famous quotations to the English language, including "Hell hath no fury like a woman scorned," also from "The Mourning Bride."
The line "Music hath charm to soothe the savage breast" is spoken at the curtain-rise of the first act and first scene by the character Almeria, princess of Granada. She believes that her secret husband, Alphonso, has been drowned in the wreck of the ship on which they were married. She doesn't realize that her father, King Manuel of Granada, has captured Alphonso and Queen Zara, who also is in love with Alphonso. In plot complications, Almeria believes Alphonso is in love with Zara, and both King Manuel and Queen Zara die in a Romeo-and-Juliet plot twist. The play was well received but, because Congreve was much better known for his comedies, it confused audiences and critics.