re: 'Take My Answer Off The Air'? page 3

It means "I will accept my answer 'off the air'." Perhaps the caller realizes that not all the listeners will be interested in hearing the answer to his question, so he tells the host "you can answer me privately; you don't have to broadcast your response to my question." It's a polite way for the caller to say "you don't need to use up more radio time on me, just tell me the answer but don't include it in the program."

That is incorrect. It means they will hang up and listen to the answer on the radio, instead of staying on the phone line to listen to the response.

I think you said it best, but it's not necessarily annoying. Sometimes in a radio forum setting it's ok to hear someone just go "Peace, respect, I'm out. Next up" after asking a good question. I could maybe see my younger self doing this out of cowardice though, ouch.

The phrase is definitely confusing. The caller almost surely isn't recording a response off the air for the speaker to pick up after the show. It's just a somewhat silly way of "handing the microphone back" to the expert and stepping away. I will say though, if someone cares about something enough to call into a radio show about it, it would be nice if they stuck around to ask follow-ups questions or maybe even offer criticism. But lobbing up a good question and ducking out is ok sometimes too.