How did Greek statues go from standing stiffly to standing at ease?

By learning to let the weight rest on one leg. Early Greek statues of nude youths, called kouroi, stand with the left leg forward as though walking, yet they could just as easily be standing still or striding. The pose takes after the style of Egyptian figures. Many were found in cemeteries, where they likely served as tombstones, so not all of them are gods.

Around 480 BC a sculptor did something new. The Kritios Boy is the first known statue to use contrapposto: a figure standing with most of its weight on one foot, so that the shoulders and arms twist off-axis from the hips and legs. The change runs through the whole body. The pelvis rises on one side, a buttock relaxes, the spine curves, the shoulder line dips to counter the pelvis, and the rib cage looks naturally expanded as if breathing.

The statue was dug up on the Athenian Acropolis, in the dump where Athenians buried sacred objects broken by the Persian army in 480 BC. It is considered a crucial development because contrapposto marks the first time in Western art that the body is used to express a psychological disposition.

Even "first" needs a caution. It is possible, even likely, that earlier bronze statues had used the technique, but if they did, they have not survived.