How does an Induction "Transmitter" work and how would I go about making one?

It depends on what you are trying to transmit, I suppose. Any old wire with a moving current in it is an “induction transmitter” for some purposes, but it sounds like you want something that can pick up sound through a mic and energize a coil sympathetically with its signal.
I can’t imagine why you would want to do this, but looking at the schematic you present and applying my infinitesimal electronics knowledge suggests that it would work.