The rule has varied origin stories, but this summarizes it:

How the Rule Got Started

In the early 1900s, the rule about only wearing white between Memorial Day and Labor Day was started according to the vacationing schedule of the elite, when there "was a dress code for practically every occasion." Those who could afford it would leave their homes behind and spend months at the sea or in the mountains, where they had a completely different type of wardrobe, including the light, bright white pieces most associated with the summer months. When those fun-loving summer days were over, the white clothes were packed away and the people went back to their regular lives, having escaped the oppressive heat between the end of May and early September. Those white clothes that they loved during the summer months had no place in their "real life" wardrobes, so the cutoff for white clothing became Labor Day.