The "g" in "0g" is not a measure of the gravitational force you're experiencing, it's a measure of the forces that are stopping you from moving freely. When you're standing on the surface of the earth, your body wants to accelerate under the influence of gravity downwards at 9.8 meters per second per second, but the surface of the Earth is pushing upwards on the soles of your shoes with sufficient force to hold you stationary. If you were standing on a spring scale (these devices measure force, not mass) it would show some non-zero force.
However, when you're skydiving there's no force stopping you from falling freely. A scale under a skydiver's feet would read zero (in fact, we'd have to fasten it to his shoes to stop it from drifting away). If this isn't completely clear, you could imagine standing on a spring scale on top of a trapdoor - as long as the trapdoor is closed the scale reads your weight, but as when it opens and let's you and the scale fall free, the reading goes to zero as the scale is no longer being squeezed between your feet and the trapdoor.
As an aside... When astronauts in orbit around the Earth experience 0g, it's not because they aren't experiencing gravity (if that were the case, there wouldn't be anything holding them in orbit), it's because they're free-falling just like the skydiver. The only difference is that their trajectory doesn't intersect the surface of the Earth while the skydiver's trajectory does (which is why he will not be happy if his parachute doesn't open to stop his free fall).