The user Perran already gave a very good answer to this question, I'm just going to go a notch further and take it to some extremes. There are really two things to consider, there's your "weight" which is the Force of gravity pulling you toward Earth and your "apparent weight" which is really how pulled down you actually feel.
Imagine floating in a lake, gravity is pulling you down the same as it would on land so your actual weight hasn't changed at all. But floating in water seems different to standing on land. It's because on land, the ground beneath your feet is being squashed - all those atoms you're standing on are being crunched together like little springs so they're pushing back up on you, providing an opposing force. Your "apparent weight" is this feeling of the floor pushing up on you. But when you get into water, you sink into it a bit and it doesn't squash up against you as much, so you don't feel as heavy.
Or imagine being in a lift. As the lift moves upward, you feel heavier, it's because the ground is pushing into you more, so your body feels more of a Force from below, making your "apparent weight" (i.e. the force between the ground and you) pretty large. If the lift were to start falling, the ground beneath you wouldn't be pushing up anymore, so you wouldn't feel yourself getting squashed against the ground, giving the apparent weight of nothing.
So when you're in free-fall, that feeling of rushing toward the ground is your weight pulling you, but there's nothing squeezing back up into your feet (no ground surface) so you feel completely weightless. In reality you still have lots of weight, that's what's making you move, but your apparent weight is essentially zero (apart from the air particles bumping into you)...and from there we can go into Perran's answer which explains terminal velocity nicely!