The Knickerbockers Base Ball Club of New York were the first organized baseball team and the founding members of the National Association of Base Ball Players (NABBP) in the late 1850s. They were also the first team to wear official uniforms, note they are not in knickers.
Taking their cue from rules being used by many less organized baseball games like "town ball," "three-out, all-out," and "one old cat," the Knickerbockers were the first to write down rules that established a standard distance between the bases, introduced foul territory, and, most importantly, ended the practice of throwing the ball at runners ("soaking" them) to get them out.
Cartwright did indeed leave New York to seek his fortunes in California, eventually going all the way to Hawaii. He introduced the Knickerbocker baseball rules to towns and players all along the way, becoming a sort of Johnny Appleseed of America's game.