The answer is that when you get close to an object, what you are actually feeling is the force created by the repulsion between the electrons in your hand (or other body part) and the electrons of that object. Different objects give you a different sensation because their unique chemical composition and physical structure causes varying amounts of forces (in the repulsion between your electrons and the electrons of that object).
The example given in the movie specifically refers to the bouncing of a basketball on a basketball court. You initially give the ball a downward velocity (which over time increases due to gravitational acceleration) and send it falling towards the concrete (or whatever substance the court floor is made of). As the ball starts approaching the floor, there is a slight repulsion between its electrons and those of the floor. As the ball gets closer, the force of repulsion becomes greater and greater (to keep the approaching ball's electrons from ever coming into physical contact with the floor's electrons). Eventually, this force becomes so great that it 'bounces' the ball back up towards you.
I just found it very interesting to imagine that at this moment, as I type on my keyboard, I am not actually touching the keys. The force of repulsion given off by the electrons on my fingertips is simply greater than that given off by the keys, which is why I am able to force them down and type this post. Kind of makes you question some of your basic perceptions on the world when you find out that something you that was so obvious or knew so much about turns out to be completely different than you had originally thought.

