When I was a little kid and a prayer was being said (at the dinner table, church, before a math test, etc.) we were told to close our eyes. I’ve always noticed that people tend to close their eyes during prayer. Even on the childrens shows on the religious channels they show the puppets covering their eyes during prayer. Any reason why?
I would guess it is to prevent any outside distractions from taking your mind off your prayer, but, as far as I am aware, there is no scriptural instruction to do it.
Just a guess, but probably the idea is to eliminate distractions and allow the person doing the praying to focus on God and spiritual things rather than on whatever bright, shiny objects or other visual distractions happen to be lying around.
This leads me to a mental picture of people praying with their eyes shut and their fingers stuck in their ears, but I suppose that would be undignified.
This is what I’ve heard also. It just helps you focus. I don’t believe there is any specific Scriptural injunction to do so, but it is natural for people to shut their eyes when concentrating, and so it follows naturally to do so when concentrating in prayer.
Of course, sometimes I say a prayer while I’m driving, and then I try not to close my eyes.
When you are talking to someone who is invisible it is good form to close your eyes, otherwise they may feel uncomfortable. Not everybody likes being invisible and by closing your eyes you make them more at ease. 
It keeps you from paying attention to the man behind the curtain. Out of my Top 10 cynical answers to this question, that’s the one I decided to type.
Another possible reason: holdover from the idea that God is so holy that if a human were to look at God, the human would die.
But I think mostly it’s about focus and respect.
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But I think mostly it’s about focus and respect.
It serves both to focus the thoughts of believers, and to force the ritual and at least the appearance of praying on others. I remember being told off at a school assembly for not having my eyes shut: but as a child was in no position to say, “Ah, but how did you know?”
At one point in traditional (orthodox) Jewish prayer, it is customary to close or shield one’s eyes, because the Divine Presence may become manifest at that moment, and one shouldn’t look upon it.
Otherwise, I’d guess it’s a way of focusing internally and not being distracted by the legs of that gorgeous woman sitting in the next pew.
Reminds me of a line from a poem that I wrote for my Confirmation:
And when I pray, I close my eyes
To let my prayer flow freely
To the world around me
Where God dwells
Says it all… though personally, I prefer to keep my eyes open!
iirc, there are many references to Jesus ‘looking’ into Heaven and praying. too lazy to check for cite.