Diagnosis
The logic board is sending the signal, but the main power is not reaching the drive motor. This is often a failed start capacitor.
How to Fix It
Locate the cylindrical start capacitor on the logic board. It is typically 150-200µF. Test it for capacitance and shorts. Replace the capacitor if it's bulging, leaking, or tests bad. Solder in a new one with the same voltage and µF rating.
If That Didn't Work
The motor's thermal overload protector may have tripped and not reset. Let the opener cool down for 30 minutes. If it still clicks, the protector is faulty and the motor may need replacement.
Heads Up
Capacitors can hold a charge. Discharge it safely with a resistor across the terminals before handling. A clicking board with no motor movement stresses the relay. Fix it quickly.