You still fall because you are not weightless, despite feeling that way. You still have mass and are still accelerated by gravity toward Earth. An object's weight is defined as the force of gravity on that object and is calculated by multiplying the mass of the object by the acceleration of gravity experienced by it,
weight W = m (mass) * g (gravity acceleration). However, weight can only be felt when there is something opposing the gravitational force such as the ground you are standing on or a chair when you are sitting. This is because acceleration of gravity is essentially the same on all parts of your body (*note below) and therefore causes all parts to accelerate at the same rate. When there is something opposing that force (basically, preventing you from falling toward the center of Earth), you are then able to feel the pull of gravity as a reaction force against you and to some extent the strain on your body as you are compressed against the unmoving ground/chair/whatever.

If you are in free-fall, then there is nothing to oppose gravity and therefore your weight cannot be perceived.

The opposite of this is when accelerating away from Earth, such as going to a higher floor in an elevator or up a hill on a roller coaster. Then, there is an additional acceleration of something (elevator floor, seat) against your body. This increases the reaction force against you, causes your body to be compressed slightly more, and your apparent weight increases (your "real" weight is still m*g though).

Here is a simple simulator to demonstrate the effect.

*This is only the case because Earth's gravity is weak and its gravitational field is practically constant over the distance of the height of a human. Black holes have much stronger gravity and the gravitational field near them is non-uniform enough that a person or object falling into one would feel a different force from gravity at their head than at their feet. This effect is called spaghettification and has been observed for a star falling into a black hole.

Great question! You still have mass and weight when falling.

Weight is the force of gravity acting upon your mass. However, you feel weightless falling because no ground is pushing back on you from beneath. This means standing on the ground, you feel a stronger force acting upon you. Gravity pulls you down, and the ground counteracts this pull. This is called a normal force acting in the opposite direction (Newton's 3rd Law of Motion). In fact skydivers feel air resistance as they fall. While they may feel weightless when they jump from a plane, they come in contact with air resistance. This force pushes back on them.

You ask a great question! I think the conundrum to resolve here is what we define as weightless.

When you are falling, you still have weight and accelerate towards the ground at a rate (9.8 m/s²) dictated by the gravitational pull of the Earth. The "feeling" of weightlessness though is when you are not being supported by something below you and are in free fall. So, when you are falling, you still have weight but you feel weightless because your body does not feel support below it and is not doing the work it typically does to keep you walking/upright on the ground.

You aren't actually weightless. Instead, gravity is acting on every atom in your body with exactly the same strength. This has the consequence that you don't feel any pressure, i.e. you're not being stretched or compressed, which is how you normally sense that a force is acting on you. Because gravity doesn't normally exert such forces, you don't feel any forces, even though you actually are being acted on by gravity.

You still weigh whatever you weigh on Earth - you just feel different. On the moon, a bathroom scale would give you a much smaller weight, but that's because the moon's gravity is so much lower than Earth's gravity.

When in free fall, the only force acting upon your body is the force of gravity - a non-contact force. Since the force of gravity cannot be felt without any other opposing forces, you would have no sensation of it. ... As such, you would experience a weightless sensation. Source.