Why do GPS satellites move in relation to earth's surface? Wouldn't it be easier with geostationary satellites? Has is to do with the monitoring of gamma-ray bursts?
Geostationary orbits (GEO) are necessarily over the equator, and satellites in GEO are not visible at all poleward of about 80 degrees -- they are always below the horizon (There's a good explanation on physics.stackexchange.com).

This is a problem because for GPS to work, you need at least four satellites to be visible and they need to be well-separated in the sky.  Because it's rare for the whole sky to be "visible" all the time -- parts are blocked by buildings, metal vehicles, hills and suchlike -- you in fact need more than four in the sky as seen from any point on Earth to have a minimum of four actually visible all the time.

So even at lower latitudes where GEO satellites are visible, they still all are in the southern (or all in the northern) sky and thus won't give a good GPS fix.  And the military (who were paying for this) wanted to be able to use GPS in the north, not just in the equatorial regions.

Also, possible GEO satellite locations trace a gently curved line across the sky, and even if there were a dozen visible and even at low latitude, they'd give very poor accuracy perpendicular to that line.

The solution was to put the GPS satellites into many different orbits which were strongly inclined to the equator. They wanted them to be fairly slow-moving as viewed from Earth, so LEO wasn't viable.  (LEO also required too many satellites to get adequate coverage.)   They needed to be above the Van Allen belts for longevity. But they also needed them to be as low as possible to maximize signal strength. 10,000 miles up turned out to be the right engineering compromise.

Gamma ray bursts were lagniappe.