Q:

Why do flat objects land softly?

I noticed that if you drop a flat surface, parallelly from a relatively low height, onto another, the impact is almost cushioned.
What is happening here?
Is this to do with the air, especially near the centre, not being able to escape, thus applying a repulsive pressure?

A:

What is happening is that there is a volume of air under the thing being dropped, which has to make its way out the sides.  As the distance gets smaller, the air pressure under the object increases while the air is escaping. 
Try a flat board, then drill some holes in the board that account for maybe 10% of its area.  Despite being 10% lighter, the board will then land much faster since you have provided a lot more area for the air to escape.
Of course in a vacuum, no difference and even a very light flat object would just slam down. 

A:

That can be explained using Fluid Mechanics and Hydraulics, because air is a fluid.
Torricelli's Law gives us the expression of the total time a fluid needs to empty a container. It is:
$$
\Delta t = \frac{2A}{a\sqrt{2g}}(\sqrt{h_1}-\sqrt{h_2})
$$
In your case, there is no container, but the paper's weight has a force by area over the fluid air, what creates pressure.
The air as a fluid needs time to displace and depressurize, what creates a force that holds the paper. If you did the same in vacuum you wouldn't have the same results.

