The short answer is that we don't have a definitive answer regarding the exact mechanisms. There is still much research and discussion about the precise mechanisms of the Great Red Spot and other anticyclones on Jupiter. Much of the current thinking is based on two different models—the quasi-geostrophic (QG) and the mid-geostrophic (IG). It's possible that the only way to resolve the difference between these two models will be to send probes into the atmosphere to obtain direct data.
But we have a fairly good consensus on some of the contributing variables.
The Great Red Spot is as tall as Earth and almost three times as wide. At its narrowest point, it is still six times the diameter of the largest hurricane ever recorded on Earth. The first question we should ask ourselves is why hurricanes on Earth don't last very long. Hurricanes feed on the energy in the moist air over warm water. Normally, hurricanes live until they either pass over cold water or land. On Jupiter, there is no cold water or land for them to fly over.
We should then ask ourselves what else is different about Jupiter that would affect the longevity of a hurricane. There are no boundary layers within the thin weather layer of Jupiter's atmosphere, which means there is very little dissipation. Jupiter rotates very quickly (resulting in a large Coriolis force). There is strong east-west shear. And the atmosphere is so thick that there is a virtually unlimited reservoir of energy.
There is broad agreement that a vortex can become stable if it is fed by shear in the bands above and below it. Computer models reflect this and explain why anticyclones like the Great Red Spot on Jupiter last so much longer than cyclones.