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.
1776, "a display;" see show (v.) + off (adv.). From 1801 as "a deliberate and ostentatious display;" in reference to the person who makes such a display, attested from 1924. The verbal phrase is first recorded 1793 as "make a conspicuous and obvious display." Noun showing-off is from 1874.