Continuity mistake: Through the top half of the ninth, when the Yankees are batting, the scoreboard indicates the Indians have 7 hits. But in the bottom of the ninth, after the Duke comes in to pitch and Willie Mays Hayes is up, the scoreboard shows the Indians have 8 hits. The only way this would be possible is if a batter before Tomlinson's flyout got a hit and was thrown out stretching. But it's a famous error of the movie that there couldn't have been other batters in the inning for Hayes to be up when he is.
Continuity mistake: When Hayes scores the game winner in the tiebreaker, the home plate umpire is not wearing his cap. The cap is back on when he shouts "Safe! Safe!"
Other mistake: Near the end of the movie, Ricky Vaughn strikes out the Yankees' best player, ending the inning. As they walk off the field, the Yankees' Third Base coach is still in the coaches' box. He has no need to be, he should be off the field.
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 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.