How does gravity exist in space?

Gravity is not something space contains or lacks; it is a property of matter. Every mass attracts every other mass, with a force that grows with the masses involved and falls off as the square of the distance between them. In Einstein's description the mechanism is geometry: a mass distorts spacetime around it, and other objects follow the straightest available path through that distorted geometry. Either way the conclusion is the same. Wherever there is mass there is gravity, and the vacuum between bodies does nothing to block it, because gravity needs no medium to cross interplanetary distances. The force is feeble compared with electromagnetism, but it is always attractive and never cancels out, so at planetary scale it wins.

The inverse-square law means the pull fades quickly but never switches off. A few hundred kilometres up, where the International Space Station flies, Earth's gravity is only modestly weaker than at sea level, which is exactly what holds the station in orbit. The Sun's grip reaches far past Neptune and keeps comets bound to it thousands of times further out than Earth. Between galaxies, gravity is the only fundamental force that still matters at scale.

Because the crew and the station fall together. An orbiting vehicle is in continuous free fall; nothing presses the astronaut against a floor, so there is no sensation of weight, and a dropped pen falls alongside its owner rather than towards the deck. The proper term is microgravity, not zero gravity, and the distinction is not pedantry: the residual accelerations from atmospheric drag, tidal differences across the structure and the crew pushing off the walls are precisely what sensitive experiments have to be isolated from.