When houses. offices and factories were damp, most people used public transport or walked, and diets were poor, it was important to keep warm. The temperature in spring is volatile (if that is the right word) and a warm day can suddenly turn into a very cold one. So the wise would say not to be tempted to leave warm clothing off just because the day starts warm. Not so important today but being old fashioned I still follow it to some extent.
I agree with the link(above) it's NOTHING to do with the Month,and everything to do with when the Hawthorn (Or MAY) blossoms.
If it's warm enough for the May to blossom,it's warm enough to leave off some clothing.