There is no definitive answer as to how this rule originated, but there are several theories. One common theory is that white is a summer color, known to reflect sunlight and keep people cool. As seasons become cooler, the white clothing gets packed away and white is not seen again until warmer weather returns, giving the impression that white should not be worn after Labor Day.
Another theory is that white was a color that represented wealth in the earlier part of the 20th century. Those who wore white were people who could afford to travel in the summer and break away from the drab dark colors of the working class. Once summer was over, the wealthy returned to wearing darker colors in heavier materials to keep warm in the fall and winter.