His eyes never stay still and keep moving like CCTV to match my movements.
It is actually just an optical effect - nothing in dragonfly's eye is moving. Its eye is composed of long tubes spread radially in different directions. The tubes that point in your direction appear darker, because you see through them to the other end (bottom of the eye), while for all the tubes that you see at an angle you see their walls that are of different color. When you move around different tubes happen to point straight at you, which gives an illusion of dark spot following your movement.
Thanks for the explanation.