The garage door opener as we know it- electrical, that is- was first created by a man named C.G. Johnson in Indiana in the twenties.
Electrical garage door openers do not actually supply the lifting power to close or open a garage door- that actually comes from the springs (counterbalance springs) that exert pressure to lift the garage door by way of cables. The electrical opener exists to control to what degree the door opens or shuts, not to actually supply the lifting power. (In modern garage doors, openers also serve as locks.)
An electric garage door opener is mostly a power unit- this contains the motor, and is attached to a track. There's a trolly, attached by and arm to the top of the garage door, that slides backwards and forwards on the track, forcing the door to open and shut. This trolly is forced along the track by one of three possible mechanisms: a chain, a rubber belt, or a metal screw that twists as the motor turns on. The trolly also has a mechanism that can quickly release the garage door for power outages or other situations in which the doors needs to be manually operated.
All of these mechanisms hang above the garage door inside the garage, with the power unit near the rear.