First and foremost summer jackets protect your skin in case of an accident. The skin on your body is your largest organ, and the most subjected to the outside world. Your skin helps regulate your temperature, get too hot and capillaries near the skins surface enlarge to cool the blood, too cold and they shrink to keep the blood centralized to your core. This is where a jacket comes in, it helps the skin work more efficiently when riding a motorcycle. Let's face it, skin wasn't designed for high speeds or riding on motorcycles.
They added the sweating part. It's not accurate, sweating is a natural action to regulate temperature. When you stop sweating in extreme heat your skin would be overwhelmed. Skin, when exposed to these conditions long enough, will also burn. We've all been a lobster at some point in our lives, haven't we?
Wearing a jacket protects the skin from the sun helping to keep the skin's surface temperature lower, opposed to being directly exposed to it. Yes, when sitting on a hot bike, on a hot day, on asphalt, you will be hotter in a jacket. No one is suggesting otherwise. Once at speed, however, a well ventilated jacket will circulate air over the skin to help maintain a more survivable temperature. This creates a cooling effect due to evaporation of the moisture from sweat and the air flowing over the surface making your skin more efficient at regulating temperature
Yes, someone reading this right now is saying, "Well, not wearing a jacket lets air do the same thing!". True, the air will move over the skin creating a cooling effect. The difference is the water from the skin is evaporating much more quickly when it's exposed directly to the sun. The skins surface temperature is higher as well. Combined, this causes you to dehydrate quicker as your skin dries out at a higher rate.
As the sweat gets wicked away by different materials in jackets, it remains close to the skin versus complete evaporation. Resulting in that wonderful cooling sensation when your shirt is soaked and nice breeze picks up.
Remember a jacket can save your skin in several ways. The most important is that it will keep it in place if you have an accident.