Why does my monitor turn off when I plug in this cable?

@livfe is correct this is a lid sensor which has two functions one to turn off the display backlight to conserve power and to enable the external display as the primary display Vs the internal which can't be seen being closed.

From your description it sounds like something has been damaged either within the sensor or the circuitry within the logic board.

Replacing the sensor is the easiest to try I would do that, here's the needed part MacBook Pro 16" Lid Angle Sensor you will need to align it by carefully rotating the magnet. But let's first see if the system responds correctly at all opening and closing the lid. If it doesn't then the logic board needs repairs from the liquid spill.