Why does falling feel like weightlessness?

It's not obvious that one should feel weightless when falling. 

Gravity affects all parts of our body equally.  When we are at rest, this downward force is balanced by an equal and opposite force in resistance, delivered by the ground.  The sensation of weight is the compression between these two forces.

In the absence of gravity, or in freefall, the force of gravity is not opposed by any reaction force, and so we do not feel any compression.