A great horseman - Bruce Sandifer sums it up like this:
"The only way for a rider to get a quiet seat and hands is through movement not stillness."
This is a huge topic with many layers, but first it begs the question - what is "a quiet seat and hands"?
The answer is a rider who moves with the horse in such a way as to appear like they are still.
Secondly, what does "movement [and] not stillness" mean?
These riders flow with every movement of the horse - like they are one body, molded together. To do that, the rider must properly align their body without relying on the reins to balance their torso. Nor do they grip, squeeze or brace their legs against the horse to balance their lower body.
They have learned to ride in a relaxed, upright posture without being stiff.
They know that stillness equals stiffness. And stiffness causes the horse to stop flowing freely forward.
They have learned to balance, not by pulling or squeezing or bracing against something. Instead, they have learned to balance by moving their mid-section in time with the horse's mid-section, thus allowing them to ride with an independent seat.
For the betterment of the horse, this is something every rider should strive to achieve.