The New Liturgical Movement blog posted a great history and explanation of why we veil our crucifixes and statues on Passion Sunday. The blog post, written in 2010 was entitled, Lenten Veils. Thanks to Gregor Kollmorgen who translated Fr Joseph Braun’s Die Liturgischen Paramente, 2nd ed., 1924 page 233 ff. Here is an excerpt:
To be distinguished from the Passion veils is the large Lenten veil, which has stayed in use here and there in Sicily and Spain, at some places in Westphalia, as well as in the cathedral of Freiburg im Breisgau. It is a cloth which is hung up during Lent at the entrance to the choir. It is most often white or violet and remains until the Litany is sung on Holy Saturday. The Congregation of Rites has declared the use of this Lenten veil to be admissible on 11 May 1878 (decr. auth. n. 3448).
Whereas according to current Roman use Crosses and images are only veiled during Passiontide, in the Middle Ages the common thing was to cover them right at the start of Lent, be it from the Terce of the Monday after the first Sunday of Lent, be it – although less frequently – already from Ash Wednesday. Here and there the veiling was even done on Septuagesima. Moreover, not only Crosses and images were withdrawn from the view of the faihtful by means of veils, but also reliquaries and chandeliers, and even evangeliaries whose covers were ornamented with pictorial representations were sometimes veiled. […]
The custom of veiling Crosses and images during Lent is apparently not of Roman, but of Gallican origin. It was already known in Gaul in the 7th century, as we can see from St. Audoenus’s († 683) biography of St. Eligius (II, 41). “Mos erat, ut diebus quadragesimae propter fulgorem auri vel nitorem gemmarum operiretur tumba (s. Eligii) velamine linteo urbane ornato holoserico”, [NLM: “It was custom that on the days of Lent the tomb (of St. Eligius) was covered with a linen veil finely ornamented in pure silk, because of the refulgence of the gold and the splendour of the gems.”] we read in the same. For Italy the custom is not attested until around the year 1000 […]. In the later Middle Ages the veiling of Crosses and images during Lent or at least Passiontide was universally common.
As material for the veils which covered Crosses, images, reliquaries etc. chiefly white linen was used in the Middle Ages. […] Coloured or painted veils for Crosses and images are encountered less commonly in the inventories. […]
The custom to hang up a veil in front of the altar during Lent is already attested in the “consuetudines” of Farfa, then soon after by Aelfric of Winchester and Lanfranc of Canterbury, and at the beginning of the 12th century by Honorius and Rupert of Deutz. Initially, it was probably only observed in cathedral, monastery and collegiate churches. In the later Middle Ages, however, we also find it in parish churches. It was perhaps least extended in Italy. In modern times, the Lenten veil fell more and more into disuse, and today it is, as said before, only rarely used. Furthermore, it mostly does not serve, as originally, to veil the altar and the priest; for this purpose it is normally not large enough any longer. Rather, it is now almost only an indication that Lent has begun.
The veil was ordinarily hung up after compline of the First Sunday of Lent and remained until after compline of the Wednesday of Holy Week. In parish churches it hung between nave and choir, and in collegiate and monastic churches between choir (presbytery) and altar. It was drawn back on Sundays, feasts of twelve or nine lessons, at funerals corpore praesente and on certain solemn occasions like e.g. holy orders, the vesting of novices and similar occasions. Only the veil of the high altar was drawn back then, however, not those of the side altars. For not infrequently, a Lenten veil was hung up in front of these, too. On ordinary days the veil was either not drawn back at all during Mass, or just for the Elevation, and here and there also between Gospel and Orate fratres. Practice in this respect was rather varied according to local custom.
As for the material, the Lenten veils, in Germany also called hunger veils, were mostly made of linen […], but there were also those made of silk. […] In the later Middle Ages, it was popular to embroider, paint or imprint the Lenten veils with scenes from sacred history, especially those of the Passion. […] The enormous Freiburg Lenten veil from the year 1612 already mentioned shows a large Crucifixion as its main image. Magnificent Lenten veils with a wealth of biblical scenes are also at Zittau and in Gurk cathedral. […]
The veiling of Crosses, images etc. during Lent and Passiontide was done because these times had the character of penance and grief, and therefore decoration in the church was deemed inappropriate. The veiling of the Crosses, moreover, may have its reason in the fact that until the 12th century the representations of the Crucifixus showed not so much the Passion of the Godman, but his Triumph on the Cross. Likewise, the great Lenten veil was doubtlessly introduced with regard to the character of grief and penance proper to Lent. The veiling of the Holy of Holies – i.e. the altar – meant in a way a partial exclusion from the cult, which was to remind clerics and laymen alike, in the time of penance, more manifestly of their sinfulness and to impel them to cultivate a truly penitent disposition.
Of course, over time other meanings were additionally attributed to some of these customs, which is easily understandable given the medieval predilection for mystical speculation. In the veiling of Crosses, images and other decoration of the church was thus symbolised the contumely, weakness and humiliation, which in the Passion of the Lord veiled, as it were, His Godhead and divine Power. The veil however, which was hung before the altar, was associated to a multiple symbolism. It was called a memory of the veil of the Old Testament, which dived the Holy of Holies from the Holy and was rent asunder at the death of the Lord. It was seen as an image of the starry heavens which separate material and spiritual world and veil from us the sight of the heavenly fatherland and the glorified Saviour. It was interpreted as the veil with which Moses covered his face, whose resplendence the people could not bear, or as the spiritual shell of the old service of the Law, which still enfolds the hearts of the Jews and prevents them from grasping the clear meaning of the Law. The taking away of the veil at Easter, then, was to signify that Christ now again stands before us in the unveiled splendour of His eternal glory, that He has opened up the heavens for us and taken away the blindness of the heart from us, which had made it impossible for us to understand the mystery of His Passion.