DIAGNOSIS That's the internal relay inside the thermostat clicking on and off. It's trying to send a signal to your HVAC equipment, but it's not getting the expected response, so it cycles. This often points to a wiring issue or a mismatch between what the Nest thinks is connected and what's actually connected.
FIX The first thing to check is your wiring setup in the Nest app. 1) On the thermostat, go to Settings > Technical Info > Power. 2) Look at the readings for Voc and Vin. If Vin is very low (like below 3.7V) or jumping around wildly, it indicates a power stealing issue. 3) The best permanent fix is to ensure you have a proper 'C' (common) wire connected to provide constant power. If your thermostat cable has an unused wire, you'll need to connect it at both the Nest base and at your furnace control board.
IF THAT DIDN'T WORK The clicking can also happen if a wire is loose or if there's a slight short. With the HVAC system power OFF, pop the display off and check each wire is securely seated under its connector on the baseplate. A wire strand touching another terminal can cause this.
HEADS UP Don't ignore this. That constant relay clicking and power searching can, over a long period, drain the Nest's internal battery and lead to it shutting down unexpectedly or losing Wi-Fi connection.