The batter got a point every time he or she hit the ball and kept batting until the stool was knocked down. Then the players changed sides. That's probably the way the Pilgrims played it, although it's not certain. By sometime in the late 17th century, though, stoolball had turned into a sport something like cricket. The batter, after hitting the ball, would run from one stool to the other, getting a point for each time he or she made the run successfully before the other player retrieved the ball.
Some authorities think that cricket was an offshoot of stoolball.