All vocal eliminators start with the assumption that the lead vocal in a recording is in the center of the stereo field - which means that it is equal amplitude and phase in both left and right channels. Subtracting one channel from the other removes this highly correlated signal while leaving the uncorrelated signals in the channels unchanged. Since kick drum and bass are often recorded at stereo center the time constant of C3 and R12 is chosen so that these signals are blocked from the non-inverting input so they pass to the final output without being zapped.
In the Vocal Zapper circuit one channel is subtracted from the other by applying them to the inverting and non-inverting inputs of opamp IC1. The Normal/Zap switch allows the vocal elimination function to be disabled by switching the J2 signal from the non-inverting opamp input (Zap) to the summing junction at the inverting input of the amp (Normal). Note that the output signal at J3 and J4 is mono in either Zap or Normal mode.
IC2 is used as a mic preamp to boost the microphone signal level so that it's comparable to the program material and this high level signal is connected to the summing junction at the inverting input of IC1.
It does. Surprisingly well for such an inexpensive device. But some recording techniques can cause problems. For instance, if the lead vocal is not at stereo center but off to one side it will not be completelely eliminated but simply attenuated. If there is very heavy reverb or other processing of the lead vocal it may change the two channels so that the lead vocal is gone but uncorrelated artifacts remain. You may still hear a greatly attenuated spooky reverb-only version of the vocal. These are relatively rare problems.
Usually, more expensive units can do more processing. There may be provisions for time shifting signals so that even off-center vocals can be zapped or more elaborate filters for allowing bass signals to pass unaffected or for trying to deal with the uncorrelated artifacts mentioned above. Assembly and dealer mark-ups increase prices.