"act of ending or announcing the end of a broadcast," 1942, from the verbal phrase in the broadcasting sense (attested by 1923); see sign (v.) + off (adv.). It was used earlier as "stop doing something" (1838) and especially in reference to a formal agreement releasing a debtor from obligation.
1953 in reference to experimental theater productions in New York City, from off (prep.) + Broadway. Even more experimental off-off-Broadway is attested by 1958.