By smothering the fire your starving it of oxygen. Without oxygen it can't burn.
If you cover the fire and take the cover off quick it reignites because it quickly interacts with the oxygen again because the heat of the oil is still around the same temperature as when it caught on fire.
If you slowly slide of the lid your increasing the time you've starved the fire of oxygen and also allowed the grease to cool slightly so when the oxygen comes back in contact with the grease at a slow rate the grease won't reignite.
The best way to deal with a fire is to prevent it from happening in the first place , make sure you know the boiling point and smoke point of what your using and use a thermometer .