Why do satellites stay in orbit around Earth?

An orbit isn't a place where gravity switches off, but a fall that continues indefinitely. The clearest way to picture it is Newton's cannonball: fire a projectile horizontally from a very tall mountain and it lands some distance away; fire it harder and it lands farther; and if the speed is high enough, the curve of its fall matches the curvature of the Earth, so the projectile never reaches the ground. That's an orbit.

For the two curves to match requires a very specific speed, which depends on altitude. In low Earth orbit, around 400 kilometers up, where the International Space Station flies, that's about 7.7 kilometers per second, roughly 27,600 kilometers per hour, which works out to one full trip around the planet every ninety minutes or so. The higher the satellite, the slower it must go and the longer it takes: at 35,786 kilometers above the equator the period is one sidereal day, which is why a geostationary satellite appears to hang motionless over the same point.

What puts a satellite into orbit isn't lifting it up, but accelerating it sideways. A rocket spends its first kilometers getting out of the dense atmosphere and the rest of the ascent tilting over until it flies almost horizontally; most of the energy goes into speed, not altitude. If a satellite were released at 400 kilometers with no horizontal velocity, it would fall to the ground within minutes.

At those altitudes there's still a trace of atmosphere. Residual drag gradually slows low satellites, which lose altitude and eventually reenter and burn up unless something boosts them back up. The International Space Station needs periodic reboost maneuvers using the engines of docked spacecraft. Above roughly a thousand kilometers the drag is so weak that objects remain for centuries, which is what turns space debris into a long-term problem.