On moonless nights, why should you look at stars from the side of your eyes rather than directly?

There are two kinds of light-detecting cells in the retina of your eyes: "rods" and "cones". The cones are designed for high light-level observing, while the rods are designed for low light-level observing. The cones are more concentrated in the center of your eye, while the rods are somewhat more spread out. So by looking just off to the side, you rely more on your rods, which allows you to see dimmer objects better. This effect is called "averted vision".

Exactly how far off-center you should look to maximize this effect varies from person to person. For most people, it's between 5 and 20 degrees away from looking straight ahead.