That air presses down on the covering sheet, and that the total force it exerts grows as the area of the sheet grows. Force per unit area is pressure — so air exerts pressure.
Set the two cases side by side. Same sheet, same weight, same stick, same plate. The only thing that changed was how much area the sheet presented to the air above it. When that area was made larger, the effort needed to lift the plate became larger. Something above the sheet must therefore be pushing down on it, and the push must be proportional to the area — which is precisely how a pressure behaves.
Folded sheet → small area → small downward force → easy to lift
Unfolded sheet → large area → large downward force → hard to lift
Force ∝ Area, so Force / Area = a constant pressure — the atmospheric pressure
Why it happens: air is not empty space. It is a real substance surrounding the plate, and it presses on every object it touches. The pressure exerted by the air around us is called the atmospheric pressure.