The reason they temporarily raise their apogee so high is that it reduces the delta-v cost of changing inclination. The cost of changing inclination is directly proportional to your speed at the moment when you do the inclination change. When you are in a highly eccentric orbit, your speed at apoapsis is much lower than your speed in a circular orbit.
Also, inclination burns can be combined with raising periapsis, which allows you to save some fuel thanks to the way they are at a 90 degree angle.
Another advantage is that it provides an easy and cheap way to deliver the spacecraft into the correct position in the stationary orbit, by controlling the orbital period of the eccentric orbit so that the periapsis will be where you want it to be (possibly after several orbits).
Because of this, the best possible way to launch from far away from the equator to a geostationary orbit is to launch to a highly eccentric orbit, at apoapsis when you intersect the equator simultaneously raise your periapsis to geostationary and fix your inclination, and then circularize at periapsis.
If this is interesting, and you don't already own kerbal space program, head over to the steam store right now.
Also, inclination burns can be combined with raising periapsis, which allows you to save some fuel thanks to the way they are at a 90 degree angle.
Another advantage is that it provides an easy and cheap way to deliver the spacecraft into the correct position in the stationary orbit, by controlling the orbital period of the eccentric orbit so that the periapsis will be where you want it to be (possibly after several orbits).
Because of this, the best possible way to launch from far away from the equator to a geostationary orbit is to launch to a highly eccentric orbit, at apoapsis when you intersect the equator simultaneously raise your periapsis to geostationary and fix your inclination, and then circularize at periapsis.
If this is interesting, and you don't already own kerbal space program, head over to the steam store right now.