That wire goes up into the ceiling, to a control box usually located under the living room Air Conditioner. I believe that's where the thermostat's 5V DC comes from, and when the thermostat closes, it operates a 12V relay in that box that sends power to the furnace control board and blower.
If your two thermostats are the same, you can swap them to see if the problem is with the thermostat or elsewhere. If the other thermostat acts the same, see if you can find the control box. Be careful opening it as it may also have 120V AC inside. You'll need a step ladder and don't want to touch a live circuit while on a step ladder.