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This is a really interesting question. My explanation will probably not explain 100% of the reason behind why we close our eyes. However two of the main factors correlated to falling asleep are the circadian rhythm and the level of melatonin in the body.

The circadian rhythm is regulated by the suprachiasmatic nucleus (SCN) and is influenced by light detected by special ganglion cells in the eye’s retina. Under bright light, molecules part of the circadian cycle are degraded or activated, creating a nice cycle that then reverses itself in darkness. In drosophila, the protein TIM is degraded under light and PER gradually increases under light. The opposite is true in darkness. In humans, the corresponding molecules are slightly more complicated, but mainly use CLOCK, PER, BMAL, and CRY.

Another factor in the human body that is regulated by light is the level of melatonin in the body. Melatonin is an important factor to falling asleep. There are melatonin supplements that people use to fall asleep, instead of using other sleep-aids. (Though there doesn’t appear to be any significant harm of using melatonin over other sleeping-aids, I would still consult a doctor before using it). The production of melatonin in the SCN begins under waning light, and is inhibited under bright light. Under intense bright light at night, melatonin levels could be seriously disrupted. When its levels decrease for the night, they tend to stay low. Levels of melatonin in the body have been proven to be crucial to many different processes in the body, such as feeding, immunity, body temperature, etc. Having the correct melatonin cycles appear to be important to normal bodily functions, and is crucial for sleep.

There might be other reasons, but because of the circadian rhythm and the level of melatonin in the body, closing your eyes would be significantly beneficial to falling asleep. I suspect that the cause (and partly function) might also involve keeping eyeballs moisturized (most people can’t really blink when they’re sleeping).