Posting the model of your garage door opener might help. But in general, the click is the relay doing its job - disconnecting power to the motor. Most likely due to the current through the motor's winding exceeding a preset cutoff setting. I have had a stripped tooth from one gear stuck to the grease on the mated gear, traveled up to the next gear, wedged between the teeth of the other two mating gears and had frozen the entire box. It would move a little back and forth (as it turned out, one rotation of the gear with the stuck tooth) but the motor would shut off because of overload if I tried to operate it.
See if you can open it again and visually inspect the rest of the gears for any debris left by the stripped one. Â