Thanks for asking, anon!
Here are two great sources that talk about chapel veils.
The short and simple answer: Here
More about the veil: Here
My personal choice to wear the veil (it is a personal choice, after all) just comes from my own admiration of the beauty of reverence. Like Our Lord in the tabernacle, like Mary—the living tabernacle—the mystery of my own person is also veiled as a reminder to myself that I am precious because I belong to Him. We veil our bodies with clothes because our bodies are so good, and they are made to glorify God. I veil myself for the same reason. I am made to glorify God, and I desire to give myself fully to live out this purpose for which I was created.