Why is the other side of the pillow cooler

This doesn't constitute biology, more chemisty. To put it short, the side that your head is touching is transferring energy (heat, specifically your body heat), making the side you are touching hot. The pillow is not dense enough (to put short, the pillow is not completely filled in as there are many oxygen, nitrogen, etc. molecules from the air that can easily pass through the pillow) to have this body heat (energy) completely fill the pillow, thus the opposite side, staying at a constant cool state since it is touching an unenergized object (the bed), is unaffected by the "warm" or energized part of the pillow, and stays cooler.

unscientifically, body heat transfers to the side your touching, giving the illusion that the opposite side is magically cold.

Because the hot part moves the cold one to the other side -_-

Because your body heat hasn't warmed that side.

its farther away from your body heat~