This phrase is used in baseball to refer to the time period during an inning between the first and second outs. It is an unofficial term and is mostly used by baseball people and broadcasters and play-by-play announcers who narrate the game for television and radio. The middle of the inning has no definite time length, and may sometimes be used if there are no outs but the commenter believes that one will occur soon. This phrase is used when someone wants to refer to a play that took place during the specified time frame.
This phrase was introduced to help those managing and playing refer to a specific part of the game. When the games began to be on radio and television, it became helpful to describe to the viewer or listener what was happening. Another way to refer to the various separations within an inning is to refer to them in thirds. For example, rather than, "The pitcher came out in the middle of the sixth inning," one would say, "The pitcher came out after 5 and one-third innings."