Factual error: Shots of the first game or two have angles of the batters' shadows facing behind home plate (or in other words the sun is around left to center field in some shots). In major league games, that would not be a possibility due to baseball rules that the sun can not be in the batter's eyes.
Continuity mistake: In the scene where Willie Mays Hays is stealing home in the final game, the first baseman throws to the catcher. The catcher catches it - with his shin and left foot lying directly across home plate. It switches to an overhead shot, and the catcher is a good five or six feet away from the plate.
Answer: He just shows up, uninvited. When he first arrives, they can't find his name on the list; he's later (comedically) removed from camp in the middle of the night when security carries him out on his bed (while commenting "This is the guy who wasn't invited to camp"). So, he's relying on his speed and charm to get him a spot on the team... which, of course, ultimately works.