The inclusion of names on baseball uniforms is a relatively recent innovation. The first MLB team to sew its players' names on the backs of their jerseys was the 1960 Chicago White Sox, owned by fabled showman and promoter Bill Veeck the guy who once sent a midget up to the plate.
Other teams were slow to follow suit. There was originally some fear that the player names would cut into stadium scorecard sales ("You can't tell the players without a scorecard!"), but this didn't really happen because the names could be hard to read from the stands.
The names were very visible, however, on TV where the viewers might be less familiar with the team and didn't have the option of buying a scorecard. So with the spread of televised games in the '70s, names on jerseys became the de facto standard, and fans seemed to embrace it.
By the rules of Major League Baseball, however, names on uniforms are still optional (in contrast to numbers, which are mandatory). Several teams have elected to leave the names off their home uniforms. I suppose the argument is that the "true" fans should know the players by sight or by uniform number. (And if they don't, they need to spend more time and money watching games!) I believe only three teams are going name-free at home this year: the Yankees, Red Sox and Giants. The Cubs, Mets and Dodgers (and possibly some others) have also done it in the recent past. However, these teams typically do include names on their road uniforms, as an aid to fans in other cities who don't have the privilege of seeing their players every day.
The only team that currently goes nameless on its home and road uniforms is the Yankees. Yankee fans may claim that they are staying true to a "long-standing tradition" (ironically, the Yankees were the first team to break tradition and add numbers to their uniform backs) or they might offer some idealistic hogwash about how "the name on the front of the jersey is more important than the name on the back." Yankee haters, on the other hand, think the no-name approach is born of arrogance: The Yankees admittedly the most successful franchise in MLB history, whether we like it or not think their players are so famous and recognizable worldwide that names would be superfluous.
More: National Baseball Hall of Fame, Baseball Almanac, MLB.com, ESPN.com