People get pale when they’re cold because your body is trying to conserve heat. Heat is nessecary to maintain homeostasis, and in a response to this, your blood vessels in your skin constrict (get smaller), so less blood goes to your skin, so that your body can redirect the blood to your core, where all the good stuff is like your organs.
Blood is what makes our skin a pinkish color, if white people did not have blood, we would look like vampires. When there is less blood in your skin, the pink color goes away and leaves a pale white.
I hope this helps!