This phrase is used in baseball to refer to a relief pitcher that specializes in pitching to left handed hitters. They may also be used to force a predominantly left-handed switch hitter to bat from the right side. In general, a pitcher is at a disadvantage if the batter is a different handedness than the pitcher. This is due to the way the ball breaks on the path toward home plate. A right handed pitcher's curveball will curve in towards a left-handed batter, allowing them to see it more clearly.
A left-handed specialist will frequently pitch very few innings, and is typically used on a case-by-case basis. This helps to extend their careers because there is less wear and tear on their arms. Another advantage not often thought about by the average fan is that most batters do not face many left pitchers because not many exist. Having quality left handed pitching available is a key component to a team's success.