In baseball, this term refers to a ball pitched down the center of the plate, or a ball hit through the middle of the field. The former is the most common usage, and it comes up when a pitcher pitches the ball right down the middle of home plate, at an ideal height for the player to hit the ball, though the latter is not always a requirement. This pitch, if it were to go past the batter, would be considered a strike. The latter reference mentioned above occurs when a batter is able to connect with the ball in such a way that it travels up the middle of the field, rather than to one side or the other, as is typical.
The phrase "down the middle" is a common term used in many sports to describe an event or action that takes place in the middle of the field, or in the middle of the action. In baseball, it refers to a ball moving down the center of the plate and a batter hitting the ball into the middle of the field. In basketball, it refers to the middle of the court and the area surrounding the basket. In football, it is the area between the hash marks, and so on with other sports. The phrase started being used in baseball because it fell in naturally with what a pitcher was attempting to do with his/her pitches.