"On Air" means while broadcasting - i.e. radio or TV. "Live" means real-time, i.e. not recorded. If it is live, the producers are unable to edit out the swearing, and unable to 'bleep' it. In other words, all the listeners heard him swear.

It means during a live broadcast. The listeners heard him at the moment he spoke (ie. the broadcast was not pre-recorded and edited).