It has been a tradition in the Church to veil the Cross and the images of the saints at the 5th week of Lent, also known as Passiontide. In other areas, just like in our shrine, veiling is done after the Evening Mass of the Lord's Supper on Maundy Thursday. The Cross is unveiled on Good Friday for the veneration of the faithful, while the saints are unveiled shortly before the Mass of the Easter vigil.
In the old liturgy, the Gospel story from John (8:59), wherein Jesus hid Himself from the rage of the Jewish authorities was read on the 5th Sunday of Lent. This reflects how Jesus' divinity was hidden during the events of His Passion and Death.
In this time of mourning, the Church veils Jesus so that the faithful may more deeply reflect on His sufferings. As the glory of the Master is hidden and veiled, the saints, His servants, should not appear too.