This is a baseball reference to a hitter who bats in the middle third of the order, though it may sometimes refer to the player immediately before or after the middle third. A batter of this type is usually a strong hitter, especially the player who bats fourth, also known as the "cleanup" hitter. It is important for players in the middle of the lineup to get on base because following the middle is the bottom third. The last two or three players up to bat are the worst hitters in the lineup. If the middle of the lineup players fail to get on base or score, it hurts the team in more than just a lack of outs. The worst players will be up to bat to start the next inning, which is obviously not good for the team.
Managers pay special attention to the middle of the order hitters because they are the workhorses of the team. They are responsible for getting lots of hits, running the bases, and scoring runs. This group is led by the most powerful hitter on the team. The middle of the order is also where a player may also be the most expendable, because major league players that can hit well are very valuable to a team. If a middle of the order hitter is not performing, it is easy to replace them, at least offensively.