Diagnosis
Capacitor failure in the DC motor controller. This is a common failure mode. The 'pop' is the capacitor venting. The hum is the motor receiving power but not the start signal.
How to Fix It
The unit must be inspected. Unplug the opener. The main control board houses two large cylindrical capacitors. Visually check for bulging tops or leaked electrolyte. This is not a user-serviceable repair. You must replace the main logic board assembly (part 41A2039 for Genie-sourced models, 41A5275 for Chamberlain-sourced).
If That Didn't Work
If capacitors appear intact, the DC motor itself may have failed. Test by manually disengaging the opener and operating the door by hand. If it moves freely, motor failure is likely.
Heads Up
Do not attempt to replace capacitors yourself unless qualified. The board contains high-voltage components. The correct replacement board depends on your unit's manufacture date.