As @Ben states, every theory of 'why' we evolve a trait is just a guess and never more than part of the story. But there are theories.
The palms of our hands are a major source of heat transfer out of our body. For bears it is the nose and pads of their feet, for dogs it is their toungue, for us a major portal for heat out of our body is the palms of the hands.
Black bears are extremely well-insulated animals, equipped with a heavy coat of fur and a thick layer of subcutaneous fat that help them maintain their body temperature as they hibernate through winter. But once spring arrives and temperatures rise, these same bears face a greater risk of overheating than of hypothermia. How do they dump heat without changing insulation layers?
Heller and Grahn discovered that bears and, in fact, nearly all mammals have built-in radiators: hairless areas of the body that feature extensive networks of veins very close to the surface of the skin.
Rabbits have them in their ears, rats have them in their tails, dogs have them in their tongues. Heat transfer with the environment overwhelmingly occurs on these relatively small patches of skin. When you look at a thermal scan of a bear, the animal is mostly indistinguishable from the background. But the pads of the bear's feet and the tip of the nose look like they're on fire.
These networks of veins, known as AVAs (arteriovenous anastomoses) seem exclusively devoted to rapid temperature management. They don't supply nutrition to the skin, and they have highly variable blood flow, ranging from negligible in cold weather to as much as 60 percent of total cardiac output during hot weather or exercise.
Given this, one could see how the palms of our hands are more likely to react to emotional situations where the heart rate elevates and sometimes we get hot under the collar. The AVAs will
I am not physiologist, but the quote above comes from an article I stumbled upon a few months ago. They use the AVAs to reduce the swelling associated with exercise and got some amazing results. I guess that's another story though.