Why does an astronaut float in a space station?

The most accurate image is Newton's cannonball. Fire horizontally from a mountain: the projectile lands farther away as its speed increases. Beyond a certain threshold, the curve of its trajectory matches the curvature of the Earth, and it never touches the ground. It falls indefinitely, never arriving.

This is exactly the situation of the International Space Station. At an altitude of about 400 km, it travels at roughly 7.7 km/s and completes an orbit in about 90 minutes. Earth's gravity still acts on it, and strongly: it is precisely this pull that curves its trajectory and keeps it from flying off in a straight line. What vanishes is not gravity, but the ground or seat that would otherwise resist it. In an elevator whose cable snapped, a passenger would float the same way during the fall.

Aboard the station, the station itself, the astronaut, and a pen released from their hand all undergo the same acceleration and therefore fall identically: nothing pushes against anything else anymore, and the sensation of weight disappears. The inner ear stops receiving any vertical reference, which causes most crews to experience space sickness during the first few days.

The technical term is microgravity rather than weightlessness: tiny residual accelerations remain, caused by drag from the thin residual atmosphere, orbit-raising maneuvers, and crew movements. These are enough to disturb the most sensitive crystallization or fluid-physics experiments.

Since gravity remains almost entirely intact, a station cannot stay up on its own: drag slowly pulls it down, and its orbit must be regularly raised using the engines of docked spacecraft.