In baseball, this term refers to the innings at the start of a game: usually innings one, two and three, but sometimes four depending on the subject matter and the speaker. The actual time at which the innings are played or how long the innings last is irrelevant in this context. The early innings are played almost exclusively by the starting nine players.
The phrase "early innings" comes from a time when players would use the first few innings to warm up and get into the flow of the game. Players did not always have time prior to games to get warmed up and prepared, so the early innings were a time when fans and others would excuse sloppy play or slowness because it was the early innings. In the modern game of baseball, such excuses can no longer be made; professional baseball players have ample time and means to prepare for the game. However, this phrase is still used as an easy reference point for the first third of a game.