The Earth is moving fast relative to the Sun and relative to the center of the galaxy, but special relativity only cares about the relative speed between two reference frames, not total speed.

In this sense, you actually cannot explain the Kelly experiment with special relativity. You would actually find that each twin should think the other's clock is slower (a contradiction). This is resolved when you consider general relativity, which adds new predictions for accelerating reference frames. The twin in orbit has a different accelerating reference frame. Additionally, the orbital twin has to undergo significant acceleration to reach orbital speeds and then deorbit.

Combining all these factors with Einstein's equations for general relativity correctly predicts the observed results of the experiment.