How is vertical acceleration (free fall) linked to the sensation of "weightlessness"?

When your feet are on the ground, the force of gravity 'grounds' you by pushing you towards the ground - extrapolated toward the center of the earth. There is an equal opposite force being applied by the ground through your body, equal to the force of gravity. Remember for every action force there is a reaction force. This is what is perceived as weight by the brain.

However, when you are in a free fall, your feet are not grounded. You are accelerating towards the earth due to gravity and there is no equal opposite force by the ground, or any other support, that would be acting against gravity. Your brain perceives this as weightlessness.