Why do we have more sweat glands in our palms?
You sweat much stronger in the palms of your hand and the soles of your feet, because they contain a much higher density of eccrine sweat glands. According to Sato et al., (see reference 1) there are something like 64 glands per cm² on your back, 108/cm² on your forearm and 600-700/cm² on the palms of your hands.

This is thought to have different reasons: First, sweating is an important way to control the body temperature.
Sweat works as a defense by anti-microbial peptide dermcidin, it is also seen as am important way to maintain the conditioning of the skin and as well as a lubricant for contact surfaces. (references 2-4)

Sweating is controlled by the sympathetic (autonomic) nervous system, and can be triggered by acetylcholine and adrenaline. This explains why you get wet palms when coming into stress.

References:


Biology of sweat glands and their disorders. I. Normal sweat gland
function
Systematic review focusing on the excretion and protection roles of
sweat in the skin.
Dermcidin: a novel human antibiotic peptide secreted by sweat
glands.
The human eccrine sweat gland: Structure, function and disorders