In our parish, since the 5th Week of Lent, this has been a common (and good!) question. Here are a few thoughts by way of response:
The scriptural basis for the veiling of statues, crucifixes and other images is found in the Gospel according to St. John (8:46-59). At the end of this passage, it is related that some of the Jewish people took up stones to cast at Jesus, who then escaped and hid himself. On one level, the veiling recalls this hiddenness.
The veiling also symbolizes the fact that Christ's divinity remained largely hidden throughout his public ministry, even during his Passion and death. Only after was his divinity made more manifest.
This custom, on a very practical level, also helps us to concentrate on the great essentials of Christ's work of redemption.
Shrouds over the sanctuary crucifix and statues...