An obsolete Intuit / Roam Data credit card reader emerged from the heap:
“Turn up the volume” suggestes where the power comes from:
They drive a LOUD, probably square-ish, audio signal through both “earphone”channels, rectify and regulate the output, and have plenty of power for the reader. The card data returns through the “mic” as another audio signal; I assume they choose an encoding well-suited for a dab of DSP decoding.
Nowadays, of course, 3.5 mm jacks are obsolete, audio data travels by Bluetooth, and you must keep a myriad of batteries charged at all times.