I suppose the presence of Red and White Queens in Through the Looking Glass suggests that sets once were in those colors, but maybe it's just that Lewis Carroll happened to have a red and white set circa 1860. I have two nice sets from around 1950, one Staunton and one Florentine, and they are black and white. Well, the blacks are jet black, but the Staunton white is stained blond wood and the Florentine white wants to look like ivory.
On the other hand, I think checkers are still predominantly red and white, and I see you can buy red and white Chess sets, as well as green, multicolor, crystal, metal, and other colors that would drive chess players crazy. (Most players wouldn't like my Florentine set with its 11th century replica statues.) Maybe the best answer is simply that Staunton black and white chess sets are most recognizable and appealing to the eye.
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So you can identify the "light squares" and the "dark squares" better. The squares on the board are not always white and black. They are light and dark. Wooden chessboards will have lighter brown and darker brown. Glass boards will sometime have transparent squares and opaque squares.
The purpose of the colors of the squares is simply to help distinguish them from each other. By alternating colors, it is easier for your eyes to see the squares.
The colors do not need to be Black & White.
In fact, there are many baords in which other colors are used.
However, it is typical to have one darker color and one lighter color so there is a contrast.
--CM
some old noble dude in thick rimmed glasses decided on that so that you and I would not stress ourselves going through the process of looking for 'appropriate' colors!