From BR Bullpen
The batter's eye (short for batter's eye screen) is a solid-colored, usually dark area beyond the center field wall of a baseball stadium, that is the visual backdrop directly in the line of sight of a batter, while facing the pitcher and awaiting a pitch. This dark surface allows the batter to see the pitched ball against a sharply contrasted and uncluttered background, as much for the batter's safety as anything. The use of a batter's background has been standard in baseball since at least the late 1800s.
The area is usually painted black or dark green or some other color dark enough to allow batters to track the flight of the white ball. If there are seats behind center field, they are painted black and not occupied during baseball games, as the "black bleachers" section is directly in front of them. If fans were allowed to sit in this section, it would create an unfair pitcher's advantage, in addition to raising the batter's exposure to danger, as it would make it more difficult for the batter to track the ball if a substantial number of fans were wearing white shirts.
One example of a batter's background is the black area in the center-field bleachers section of Yankee Stadium, known simply as The Black. At one time, there were seats where the black area is now, but because of distractions the seats were removed and the area painted black. (Before the stadium's mid-1970s renovation, a batter's eye screen was often put up in front of the section.)
At Fenway Park in Boston, the batter's eye is made by laying a large green tarp over a section of the center field bleachers. During day games, the seats will not be sold so the tarp can be laid down; however, during night games, when the batter is more likely to be able to see the ball regardless of the backdrop, the seats are sold. This has often created unusual seating arrangements during night games that are made up during part of a day-night doubleheader as the sections remain uncovered for the people who have purchased the seats. The Red Sox have solved this problem by handing out T-shirts of the same color to these fans to wear during the game.
At Wrigley Field, the centerfield bleachers used to be closed off and covered by a tarpaulin, and later by juniper plants. There is now a shaded luxury suite there referred to by the Cubs as the "Batter's Eye Suite".
Some stadiums have rotating billboards in this area. In this case, advertisements are displayed in between innings, while a black surface is rotated in while the game is in play. This method is used at Shea Stadium.