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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