Why do stars disappear when I look directly at them?

Stars disappear when you look directly at them because of the anatomy of the photoreceptors in your retina.

We all have two types of light-sensing cells in our eyes, the rods and the cones. Cones see fine detail and color. Rods see better in dim light. When you look right at something that is small or far away, the image falls on a part of your retina where there are only cones. This means that if you're in a well-lit environment, you will see this object very well. If however you are in dim light, you'll see the object better out of your peripheral vision (looking just off to the side of your target) because then the image will fall on the part of your retina that has rods, which can see in dim light. This is true of everyone's eyes, but many people have never noticed it. There are a few VERY rare conditions that can exaggerate this phenomenon, but they are like 1 in 10,000 level rare. A dilated eye exam could detect them.