Recall Newton's First Law of Motion in which we come across a term "inertia".A body will remain in the state of rest unless acted by an external force or will remain in the state of motion unless external force is applied.This inherent property of object of not changing its state is defined as inertia.Inertia is of two types:Inertia of Rest & Inertia of Motion.
The coin and cardboard case is an illustration of inertia of rest.Initially coin and cardboard are at rest.Now an external force is applied on the cardboard;the coin at that instant is still.When the cardboard moves the coin,due to is inertia of rest,remains at rest and once the cardboard is fully removed it falls down inside the glass due to gravity.