Why veil the tabernacle? The immediate and short answer is, because the tabernacle in the desert had veils and because there was a veil sheilding the Holy of Holies in the Jerusalem Temple. The fact that this Temple veil was torn in two does not mean that we should abandon the use of veils and curtains. That would be a simplistic, not to say childish, interpretation of that tremendous event.
In Catholic understanding, the purpose of the veil is not to hide something, but to disclose something. When we veil the tabernacle we are saying something about its contents. Usually, the veils we use are actually curtains. This can be taken to express the fact that the Temple curtain was torn in two; access to God is granted through the death of Jesus Christ. However, we cannot see the face of God and we cannot see the Risen glory of Christ. We "see" Christ and receive the Christian Mysteries in faith, and the veil represents our faith as well as speaking of the Mysteries that we cannot see clearly in this life. These Mysteries are so great, so wonderful, so beyond our comprehension and our ability to focus, that we veil them to remind us of their magnificence and holiness.
In the Instructions of the Roman Missal we are told that the tabernacle should be veiled unless the doors are designed to be used without a veil. In other words, if veils are not used, the doors of our tabernacles must be beautifully designed and made, so that they proclaim the greatness of the Sacramental Presence. In the Old Rite the point was made that the veil is used to proclaim the Presence, as is the Sanctuary light. It has to be said that some priests apparently took down tabernacle curtains without realising their importance, and often, scratched, stained and plain-looking doors were exposed which tended to suggest that the contents are not as important as they were thought to be! It may seem strange, but this is how some people have interpreted some of the things that were done in our churches following the liturgical changes said to have come from Vatican 11. Though tabernacle curtains and veils are no longer obligatory, unless the tabernacles themselves - or their doors - are particularly well-designed, veils or curtains should be used.
I remember recently visiting Lisieux and being in the magnificent Parish Church (not the Basilica!) in the town, and looking for the Real Presence. There is a tabernacle set into the old reredos. It has no veil or curtains on it and there is no Sanctuary light. I found another tabernacle in a side altar - again, no curtains and no light. I walked down to the front of the church and as I came to the right side of the transept, there behind my shoulder was another tabernacle, set into the wall. Again, no veils or curtains, but this time there was a light. This brings up the subject of the moving of tabernacles and the Sacramental Presence, but that is for another post.