Baseball age is a player's age on June 30 of the year in question.
June 30 was picked because it is approximately the middle of each major league baseball season, and thus baseball age reflects the age that a player is for the majority of the season. It allows observers to use only one age for a player for each season.
Baseball-reference.com utilizes baseball age when it gives the age of ballplayers on its statistical pages.
The expression baseball age is also used to refer to the age a baseball player claims to be, as opposed to his real age. It was quite common until the 1960s, and until 2001 for Latin American players, for players to claim to be younger than they really were in order to appear to be more interesting prospects. It was a well known practice, leading to sentences like: "Joe Blow is listed as 25, but that's his baseball age. He looks more like 28 to me". Most of these players' true ages were found out after they retired, and their records have been corrected accordingly; however, contemporary sources will list a different birth date for them than the one now used in Baseball-reference or here.
Ron LeFlore is a recent case of such a discrepancy; when he was playing, he claimed to be four years younger than his actual age. Baseball cards and other sources from his playing days will list his younger baseball age, not his true birth date.