On re-reading, this may not be the actual answer… if it's bluetooth, then it's bluetooth - but if it works in Airplane mode, then read on...
Simply put, they work by induction.
They actually 'listen' to the magnetic field in the phone's speaker itself.
If you pass an alternating current through a wire wrapped round a magnet, the current will induce a magnetic field, causing the magnet to move.
If you attach that magnet to a paper or plastic cone - you have a speaker.
Conversely, if you vibrate that cone, you induce a current in the wire coil - & there you have a microphone.
The magnetic field generated by the speaker coil is very low, but is just strong enough that if you put a 'receiver' consisting of another coil & magnet, you can generate just enough current to power a small amplifier, which can then re-transmit that to a new speaker as amplified sound.
[There must be some kind of shielding or suppression circuit in there to prevent it 'hearing' itself, but that's above my expertise-level]
You can test this theory very simply…
Put your iPhone in Airplane mode & test it on the speaker.
The phone is not transmitting anything by radio, so it must be induction.