This is a baseball reference to the middle three innings of a nine inning game. If the game is shorter than nine innings, as it is in some youth leagues, there are no "middle innings," only a first and second half. The middle innings are no different than any other set of innings. This phrase is used when someone wants to summarize what happens during the middle of the game.
Although the phrase may sometimes be interpreted generally to mean the middle of the game, it is generally only used to refer to innings four, five, and six. For example, a sports anchor or broadcaster may report the game and say, "his pitching really got better in the middle innings, but took a turn for the worse in the last third of the game." Another way to refer to the middle innings is the, "middle third."