What happens at the ISS is that, although it still feels the gravitational pull of the Earth, its linear velocity is high enough so that it does not fall into the Earth, and keeps "orbiting" around it. The same happens for the astronauts who are within, as they are moving practically parallel to each other.
For example, if you shoot a bullet, it will eventually fall
But considering the Earth is round, if you shoot it fast enough it will eventually "miss" the ground, and it will keep trying to fall, but in a perpetual manner
And this is when it will start orbiting the Earth, with enough linear velocity. As both the ISS and the astronauts are moving this way, they will feel like they are with zero gravity, but what is actually happening is this feeling of "perpetual" fall.