Everything is moving. We typically specify how things are moving compared to something else. Geosynchronous orbit is far from the Earth and moving very quickly.
Earth rotation is very stable. So satellites do not need to change speed to match changes in Earth’s speed. But satellites do slow down a little over time, so they have engines (rockets) to speed them up slightly from time to time.
Just remember that you’re standing on a planet that’s evolving
And revolving at 900 miles an hour.
It’s orbiting at 19 miles a second, so it’s reckoned,
The sun that is the source of all our power.
Now the sun, and you and me, and all the stars that we can see,
Are moving at a million miles a day,
In the outer spiral arm, at 40, 000 miles an hour,
Of a galaxy we call the Milky Way.
It’s moving really quickly
The surface of the Earth at the equator is moving about 1000 mph but the air and everything on the surface are moving the same speed so you don’t notice
Since geostationary satellites are far away (22,000 miles) they have to cover a longer distance every day so geostationary satellites are moving at about 6900 mph to cover the extra distance
Since both the surface and the satellite go around once every 24 hours it appears to hover in the same spot all the time
Motion is relative. In Earth’s reference frame, which spins once per day, they are not moving, but that’s not an [inertial reference frame](https://en.wikipedia.org/wiki/Inertial_frame_of_reference). In a non-rotating reference frame, they’re moving about 6700 miles per hour.
The way orbital mechanics works, the lower an orbit is, the faster it is. At the altitude of the International Space Station, it takes aout 90 minutes to go all the way around the world once. At the altitude of the moon, it takes about 27 days. At about 22,000 miles up, the time it takes to go around the Earth is exactly 1 day, which means the satellites at that orbit spin around the Earth at exactly the same rate that the Earth spins around its axis. That’s how satellites at that height always stay above the same spot.
Nobody has yet fully addressed the second half of the question: “How is it able to keep an exact match with earths rotation?”
The orbital height and speed of a geosynchronous orbit are unique, and determined by Earth’s gravity (and Kepler’s laws of orbital motion). In theory, if you put something in the right place and the right speed, it will stay there forever.
But in reality such satellites tend to drift a bit, partly because there’s no such thing as zero error and partly because Earth is not perfectly round. Likely the Moon also has an effect, I dunno.
Therefore a geosynchronous satellite typically has little jets that keep it in place with tiny pushes from time to time. And when a satellite becomes obsolete, the last act of these same jets is to push it out of geosynch, to reduce crowding.
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