Another reason, is that it is an act of humility. When you are in the presence of an earthly king, you do certain things to show that you recognize their power and authority. So, when we are in the presence of the King of Kings, we should do even more. For example, when men go into a church, they should take off their hats, revealing their balding heads, which is an act of humility. Likewise, a woman's hair is her glory, and should be covered, to show humility.
It also shows our longing to be the bride of Jesus when He enters into our bodies in the Eucharist.
Finally, it is a physical reminder of where we are, and Who we're with. I find it helps me to concentrate more fully on Jesus and the Mass when I veil.
I first started veiling just after this last Christmas. At first, I felt awkward and embarrased when I would go to daily Mass at other parishes, where I would be the only one veiling. I even went to one parish where people were staring, pointing, and whispering about it. Finally, I decided that God had called me to veil for a reason, and that was just between us. If people thought I was weird or crazy because I wanted to humble myself before God, then that was their problem. I still sometimes have twinges of awkardness now, when I go to Mass at different places, but mostly I'm fine with it. I like to think that when I veil at other places, it will remind others around me who may point or stare, that we are in the presence of God, and that maybe they should try to be more reverent too. Of course, I'm not saying that I'm perfect, or that I never get distracted at Mass. Even when I get to go to Mass without kids (very rarely), my ADD frequently gets the better of me, and I start looking around, or thinking of other things. But then, my veil catches my eye, or it starts to slip, or itch, and I remember to focus my body and soul back on God.
So, that's why I've started (and will forever more) veil in the presence of Jesus in the Eucharist. Maybe you will think about it, pray about it, and start veiling too! Give it a try, I think you will soon see the beauty in it too. And, if you want a nice veil, I got my first one at http://veilsbylily.com/ (I think the average price on there was about $25 plus shipping). I really like mine, but I love the way the one I made for my mom came out. (See my first post, "Welcome!") Let me know if you want me to make one for you!
I also love how my veil now smells forever like incense! :-D