Ever since baseball became organized in the early 1900s, Major League Baseball has been exempt from the Sherman Antitrust Act. Without boring you, the dispute began because a separate league known as the “Federal League” tried convincing players from the MLB to play in their league, but that came with little success. When the Federal League took the MLB to the Supreme Court, the court sided with Major League Baseball mainly because they saw baseball as simply a game; not a business that makes money.
All professional sports are essentially monopolies. A random person cannot just start a team that competes in the AL East; the owners have to vote on it and after a long process, the random person would most likely get denied.