When in free fall, the water and the cup both experience the same acceleration. Therefore, they both move together. Therefore, there is no "force" that wants to separate the water from the cup.
how weightlessness comes into the picture?
OK, forget the cup and forget the water. Let's consider a bathroom scale instead.
When you stand on a bathroom scale it measures your "weight". That is, it measures the force between the soles of your feet and the ground. Your body wants to "freely fall" toward the center of the Earth, but the ground gets in the way: The ground pushes up on your feet exactly as hard as is necessary to stop you from falling, and the scale measures that force.
Here's what happens if we move both you and the scale to the International Space Station. Your body doesn't just want to freely fall toward the center of the Earth, it actually does freely fall. And, the scale also freely falls, and the space station surrounding you also freely falls. There's nothing up there to push up on your feet to stop you from falling because everything is falling together. Even if your feet are touching the scale, and the scale is touching the wall, there's no force acting because there's nothing that gets in the way of anything else falling.
Weight is force. No force is no weight... Weightlessness.