Automatic transmission clutch packs operate generally on the same principle as the clutch in a manual transmission. When fluid pressure is applied to a servo, these clutch packs are pressed against each other. This compression of friction material against friction material is what causes the transmission to create output torque. Clutch packs will be found in a drum like container with keyways in the top that teeth in the clutch packs fit into. Alternating clutches in this drum will be splined on the inside. When pressure is applied to the solenoid and the clutches are compressed, torque is transferred to the output shaft in a similar way to how torque is transferred to the input shaft in a manual transmission.
For more info, see Automatic Cars: Which Make and Model Fits Your Needs?