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Can somebody please explain this to me. How is it possible the cap didn't fall since there's air trapped inside the container? If not mistaken, the trapped air has a pressure same as the atmospheric pressure right? So the pressure exerted by the trapped air and the surrounding air pressure should be equal, which is atmospheric pressure. Thus, the net force acting on the water is only its weight. As a result, the cap should fall down along with the water due to the gravity...But why is this happening ?

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You premise is wrong. The air inside the inverted jar is not at atmospheric pressure. It is well below atmospheric pressure. The water is acting as a piston which is pulling down with a force of mg. That force is being spread over the cross-section area of the jar and reduces the pressure in the air above it.

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