For most American teens, the arrival of the Beatles’ “Rubber Soul” 50 years ago was unsettling. Instead of cheerleading for love, the album’s songs held cryptic messages about thinking for yourself, the hypnotic power of women, something called “getting high” and bedding down with the opposite sex. Clearly, growing up wasn’t going to be easy.

Released in the U.K. on Dec. 3, 1965, and in the U.S. three days later, “Rubber Soul” remains one of the most critically acclaimed albums in rock history, influencing a generation of...