A crumpled piece of paper presents less surface area to be acted upon by air resistance than an uncrumpled piece. Both pieces are subject to the same downward acceleration caused by Earth's gravity, but the uncrumpled sheet of paper has a larger surface area in contact with the air and thus encounters more resistance than the balled-up paper.
Acceleration due to gravity is essentially constant across all objects on Earth. Gravity accelerates everything at a constant rate, which on Earth is 9.8 m/s/s, regardless of mass. This means that in a vacuum or near-vacuum, such as on the moon, lead weights and feathers drop at identical rates, being subject to identical acceleration with no mitigating forces.
Earth, however, has an atmosphere that provides resistance to falling objects. While the strength of this resistance is influenced by multiple factors, the two most important are the falling object's speed and its cross-sectional area. Air resistance is caused by molecules of air striking the surface of the falling object, so in this experiment, it follows that the more molecules the paper encounters, the stronger the resistance to its fall. An uncrumpled sheet of paper has an enormous cross-section relative to a crumpled ball and so, using the same principle as a parachute, it encounters more air molecules on its way down, slowing its fall.