"The device he came up with consisted of an antenna and a cylinder with a thin membrane that acted as a microphone. Soviet agents stationed across the street from Spaso House would turn the device “on” by focusing a radio signal on it, which then bounced back to their radio receiver. When the ambassador or anyone else in the study spoke, the sound waves caused the membrane to resonate and alter the signal that returned to the Soviets, allowing them to hear the conversation."
The wikipedia link in a sibling comment has even more information.
It could contain more detail but sounds like it was altering a radio signal that was completely external to the device and therefore had no battery at all.
“It had no power pack of its own, no wires that could be discovered, no batteries to wear out...”
this type or degree of "vibrating resonance" is passive "signal electronics" rather than the active power electronics inside RFID, which uses a stronger signal resonance to generate power to power up an active broadcast and computing circuit.
it's not entirely unlike bouncing a laser off the outside of a window to detect the conversation inside, though that higher tech and frequencies means the system does not need to be tuned to a resonant frequency.
Starts at 3:32 -- https://youtu.be/mAai6dRAtFo?t=212