Final answer:
Astronauts aboard the space station experience weightlessness because they are in free-fall, in an orbit around Earth. As they fall towards Earth, the curve of the Earth falls away, making them feel as if they are floating.
Step-by-step explanation:
Astronauts experience weightlessness aboard the space station due to them being in a state of free-fall. They are in orbit around the Earth, continuously falling towards the planet but at the same time moving forward fast enough that the curve of the Earth falls away beneath them. Thus, while they are indeed under the influence of Earth's gravity, they do not feel its effects as they would on the surface because there is no underlying support force. Instead, they fall along with their spacecraft and any other objects, so they appear to be floating relative to their environment.
This sensation is similar to what would be experienced by passengers in an elevator with a broken cable or aboard an aircraft executing a parabolic flight path to simulate zero-g conditions. The astronauts, however, are actually in a stable orbit, not falling to Earth but instead continuously falling around it due to their tangential velocity matching the rate at which Earth's surface curves away from them.