I quite agree that it is not intuitive. However, orbital mechanics are frequently not intuitive, probably because we don't get to experience an orbital environment on a regular basis (if ever).
Let's just assume we're talking about circular orbits for the remainder of my post, since you are a beginner in orbital mechanics.
There is only one speed that a given circular orbit of a certain altitude can go. Keep in mind that stable orbits do not require any force from an engine to keep going as they have been. Basically, in a circular orbit, the falling-toward-the-planet motion is matched exactly by the moving-forward motion.
Sir Issac Newton figured this out, and exemplified it with a thought experiment called Newton's Cannonball.
Note that if the orbital speed is too slow for that altitude, the cannonball crashed into the planet.
And if the orbital speed is too high for the altitude, the orbit will be an ellipse, rather than circular, or the cannonball may even escape Earth altogether!
Finally, if the cannonball is launched at the 'correct' orbital speed to be in a circular orbit at that altitude, it will neither crash, nor fly away, but will remain stable, traveling around earth at that particular velocity.
At different altitudes, this Goldilocks velocity is different. If the orbit is closer to the planet, the effect of gravity is higher, so the orbiting object must be moving faster to counteract the falling. When the orbiting object is further away, there is less falling force due to gravity (because gravitational force is based on distance), and so the object does not need to be moving as fast to counteract the falling force.
From Wikipedia's Geocentric Orbit article, we know that Low Earth Orbit could be, for example, an altitude of 160km. At this altitude, the Goldilocks velocity to keep a circular orbit is about 8000 m/s, and takes about 90 minutes.
Now what happens if we look at a slightly higher altitude? Well the velocity is lower, and the path the orbiting object travels gets bigger (the circle is bigger), so both of those factors make the orbit take longer. A slightly higher orbit might take 100 minutes instead of 90.
For a geosynchronous orbit, the orbit has to take 24 hours instead of 90 minutes, because the earth takes 24 hours to spin. This happens when the circle is expanded to an altitude of about 35000 km. The Goldilocks velocity at this altitude is about 3000 m/s.
This is all somewhat simplified, but the broad strokes are all there. As Organic Marble pointed out, you could try to force a craft to orbit at a different altitude in a 24 hour period, but it would not be a stable orbit, you would need engines to keep it going.