Meanings to do with exposing to the air in various senses date from the time of Shakespeare, and the now familiar use with opinions, grievances, etc. as objects date from the late 19th century. The meaning 'to broadcast' (first recorded in the mid-20th century) draws on the sense of the noun 'a medium for transmission of radio waves'; this originated in American English
(It aired a heartwarming TV commercial on the importance of savings institutions—Wall Street Journal, 1989)
and has become common also in British English in transitive and intransitive uses
(The obligation to keep records of all programmes aired—Economist, 1981
the controversial docudrama Bloody Sunday, which caused a storm when it aired on British TV last year—film review, British English 2004 [Old English (up to 1150)C]).