How do the three wires for each HDMI Channel relate to a single signal?
The cable and signalling technique you describe is for balanced differential data transmission. It's unlike a single output and ground that would feed via coaxial cable.

To receive a transmission you use a differential receiver. To transmit you use a differential line driver. This applies to both data and clock signals.

This technique emits less EM  radiation and consequently is less susceptible to EM interference. It also relies on a balanced resistive terminator to impedance match to the cable's characteristic impedance.