The softness of grass makes it relatively easy on the legs (except when the player slips), and its shorter points mean less running. Shorter points also mitigate somewhat the stress on the arm, as the faster and more frequently off-center ball impacts are at least fewer. Racquet also meets ball generally lower on grass, and on groundstrokes, meeting the ball low usually strains the arm less than meeting it high. When a player does slip, grass cushions the fall, especially when it's still untrampled.
If you've watched Wimbledon or the handful of other grass tournaments over the years, you've probably noticed seeing more serve-and-volley tennis there than anywhere else. Low bounces make getting under the ball to hit topspin passing shots more difficult, and unpredictable bounces add an incentive to hit the ball in the air; therefore, volleying becomes especially advantageous. Slice groundstrokes are also rewarded on grass, as it enhances their low bounces. Playing on grass thus encourages a versatile, all-court game.
Grass is the closest thing to a fountain of youth for balls and shoes. They don't get to keep their good looks while enjoying longevity, but given a choice between green coloration and being entirely used up, I'd go green every time. Earth itself agrees.