I don’t think they’re joking at all… what they said is true. If the phone has an FM receiver then the wired headphones do serve as the antenna for it. It will not work without wired headphones.
Back in the day that was a good way to have free music over the air if you didn’t have an unlimited data plan, which most people didn’t.
It’s just obsolete tech now. Having a headphone jack doesn’t mean anything if the phone doesn’t have an FM receiver. Most phones don’t have an FM chip in them anymore.