During free fall we push against nothing (and nothing pushes against us) and therefore we do not feel heavy.

You can understand and "see" this (at least in outlines) with the help of an elevator. If you step on scales in an elevator and drive downwards, the scales show a bit less of your weight for a short moment - exactly in the short period of time, when the elevator is accelerated slightly.

During free fall the scales would show simply zero, because you cannot push against the scales and the scales cannot push against the ground.