In modern day we are taught the best way to deal with heat is to remove as much clothing as possible. This exposes the skin to air, but also to sun. This is a 20th century change for most societies. Prior to this people continued to stay covered in the summers, not due to modesty reasons, but because it really does help with the heat and sun. No matter your skin color, white fabric will always be lighter in color and therefore absorb less heat than your naked skin will. If you think of a chicken in the oven, the skin of it will burn and crisp if not covered up by foil. A single, thin layer of white fabric can make all the difference on a persons skin in the same way. Add in the lost fashion for hats and parasols as well.

Today, most of our clothing contains polyester, rayon, spandex, and other man-made fibers. Most of these are plastic in nature and breath just as well as you would expect plastic to. Even a hundred years ago we were still using mostly natural fibers. Linen, cotton, and even wool breath easily and allow for body moisture to be dealt with, as well as movement of air. Linen does exceptionally well, being naturally inclined to wick moisture away and evaporate it even in the most humid of climates (usually even better than our skin does naturally). So, keeping covered by fabric doesn't make a person a great deal hotter and would keep them well protected against the sun.

Still today in the Middle East and other desert climates you'll find the inhabitants wearing loose robes that cover most of the body, often in linen. As for the regions that are traditionally wearing very little clothing in the great heat and sun, like areas of Africa, someone with knowledge of that regions proclivity for skin issues will have to chime in.

The post by /u/colevintage is spot on. I answered a similar question awhile back regarding how sailors dealt with sunburn, and the answer is much the same -- wearing long sleeves, long pants and wide hats in sunny climates, so as to avoid getting sunburn in the first place.