Let's start with the simple case. When there is no air resistance, for instance when your ball is falling in a vacuum, then it will fall at the same rate whatever it's mass. If you drop a heavy and a light ball they will end up splat on the ground at the same time - exactly - not a nanosecond worth of difference. And this is because of how Hassan explained it to be. When the earth pulls something down, it pulls down proportional to the weight of the body. So heavy bodies get pulled down more than light bodies, and so they fall down at the same rate.
Now let's inject some air into the scenario. Now we have air resistance. And what affects this air resistance? Three things: the type of surface, the surface area and the speed. As far as types of surfaces go, the rougher the surface the more air resistance there will be - with some subtle qualifiers I won't mention. The other thing is the total surface area. That is why a flat sheet of paper falls down slower than a ball of paper. The third thing is the speed. The faster that a body is moving through air, the more air resistance it feels. This is half the reason cars have top speeds. The faster they are going, the more the air resistance and the harder the engine has to work to push them against this resistance. On one side the resistance is increasing as your speed goes up. On the other side, your power is not increasing past a maximum. Eventually, the resistance is equal to the resistance power and you you what physicists call 'terminal velocity'. Terminal in the sense that you can't do better than it. You can't go faster without more power. This is not the reason that lighter bodies fall faster than heavier one.
Let's talk about what happens to the rate now. Does the heavier ball fall down faster than the lighter one? Yes, indeed it does. But why so? The secret is the speed. And Newton's second law. The world's most famous equation is E = mc^2. I bet if it was F = ma instead, Hassan might not have made his mistake.
Let me explain. When I say air resistance, I am talking about the force on the ball that slows it down. Imagine not dropping the two balls from the tower of Pisa. Rather, throwing them down at exactly the same speed. Now the force due to this friction at this starting moment will be same. Right? The speeds are the same and we have made the surface type and surface are the same for both balls. Then there is the force of gravity- the earth pulling down upon the balls. We know that the earth pulls down on the heavier balls more than light balls. So, then the total force that we have on heavy ball is more than on the light ball.
Now the subtle point. When we had no air resistance, the greater mass of the heavy ball compensated exactly for greater force (then just because of gravity) and we had the same acceleration. Now, this does not happen. The greater mass does not compensate for the greater force exactly. Rather, it undercompensates. And so the acceleration of the heavier body is more and so it falls faster and ends up at the bottom of the Tower of Pisa before the lighter ball.
Which is really the real explanation for why the heavier ball falls faster. The balls do not have to reach terminal velocity. They just have to fall and the interplay of air resistance, gravity and masses will be such that the heavy ball will fall faster.
(For the technical minded, the solution to the differential equation of a body falling under gravity with air resistance is determined by two parameters: b, which accounts for the resistance; and m, which accounts for the force of gravity. The solution is a complicated expression involving exponentials which is nevertheless a strictly increasing function of m.)