When we are tired, our tear ducts produce less fluid, which dries out our eyes. If we are really tired, we may drift off mentally, causing us to stare. Having our eyes open for prolonged periods of time increases dryness.
When we rub our eyes, we stimulate the tear ducts, which spreads moisture over our eyes. This pressure also triggers a reduction in blood pressure, which helps us relax and get ready for sleep.
This is the same reason we rub our eyes when we're under a lot of stress.