Exactly the same combination of falling and moving sideways works for anything in orbit. The planet, moon, or satellite falls towards the things that it's orbiting. That's the effect of gravity. But it also travels forwards, at just the right speed, so that it the sideways movement compensates for the movement caused by the falling. So the orbiting thing stays the same distance away from what it's orbiting around.
Thinks about trying to achieve this balance artificially, which is what we do every time we launch an artificial satellite. You can imagine this in two steps: one, use a rocket to get a satellite to the planned height; two, fire some thrusters to set it going it sideways.
If the satellite is
thrown sideways too slowly it will fall to Earth because the pull of gravity is too great. If the satellite is thrown sideways too fast it will escape from the Earth's orbit because the gravitational pull is not sufficient to provide the required centripetal force. With the correct
launch speed the satellite continues in its falling orbit around the Earth.
You have to set the horizontal speed of the satellite so that the gravitational pull of the Earth tugs it round on its orbital path.
When talking about planets and moons with children it is quite likely that someone will pose the (very good) question:
Abi: Miss, what keeps the moon going?
The short answer to the question is:
Teacher: Nothing keeps it going, it keeps going itself.
As the satellite is launched from the carrier rocket, a rocket thrust acts to throw it out in the desired direction at the prescribed speed. The crucial point to understand here is that the satellite speeds up only for as long as the rocket thrust is acting. Once the rocket motor is switched off the satellite continues at the final speed achieved, neither speeding up nor slowing down, and the gravitational pull of the Earth continuously tugs the satellite in and along its orbital path. In this sense, the satellite
just keeps going itself.
If the satellite was moving through empty space it would stay in its orbit forever, there being no forces acting to speed it up or to slow it down. In reality low orbit Earth satellites are not travelling through empty space and so experience a resistive force or
drag due to the
thin atmosphere which they encounter. In such circumstances, occasional rocket thrusts are needed to maintain the motion of the satellite, otherwise it will fall to Earth.
There are several ideas about how natural satellites in the universe (these could be moons, planets, stars, or even galaxies) got to start moving sideways as well as falling. These rely on matter that's already swirling coming together to make up the orbiting systems. And once stuff is swirling, or spinning, it's rather hard to stop, as you'll know if you have ever tried ice skating. IN any case, it's as well that there are orbits, with the fine balance between falling and moving sideways, otherwise we would not have a home, as Earth would not be.