Typically in this kind of setup you have multiple mics (wireless, podium, etc.) wired into a soundboard that goes out to a recording / live feed. If someone accidentally bumped (or intentionally moved) the sound board volume fader for a mic input it could mix that sound into a recording. Another way was it was picked up by Joe's podium mic but I think this is less likely depending on the mic used. I would assume a podium mic wouldn't be broadly omnidirectional which would allow people around it to be picked up and not just the one in front of it. I'm not an audio engineer, but I believe these kind of mics would be pretty focused to avoid picking up stray sounds.

edit: I know belive the most likely source was from an AV studio headset mic, where family/lawyers/ etc were watching.