Even if you watch someone as "motionless" as Rubinstein, you will notice subtle shifts of weight and signs of leaning one way or another, even in old age when he stayed closer to the keys because of failing eyesight. I used to "sit up straight" at a "perfect" immovable angle, and unlocking this in a good way is what my teacher and I are working on right now. Like in 1piano4joe's description, we actually need to involve our whole bodies in playing. Even if you appear to move very little, if a slight shift forward and back, side to side, pivoting a bit from left to right happen, supported from the feet up, your movement gets more fluid. You also have joints in more than your fingers: wrist, elbows, shoulders, and more.
I've only been at this a few months. When your body is able to adjust naturally, then the hand and fingers get looser. Or put another way, if you are tight in your body, it goes into the fingers too. Conversely, with loose wrists, the shoulders and elbows unlock (for me). When you balance in the lower part of your body as well, this helps relax a tight back, which in turn loosens the entire arm. Then you can also angle your hand at the wrist and find a good angle to get at the keys. That includes the slant of your hand on the piano keys, where your arms are, and where your body is.
I think little kids have to lean this way and that more because of their size, and this stays in micromovements when they get bigger, so that they are not sitting ramrod straight. A lot of "classical" pianists sit rather still, but if you look closely you'll see that they don't "sit up straight" like students. Something else is going on.