The basilica has been called St. Mary at the Crib (ad Praesepem) since the time of Pope Theodore I (642-649), a native of Jerusalem, and the wood is reported to have come to Rome in that century. The basilica itself says that recent scientific studies date the wooden pieces to the time of Jesus' birth, without giving details; that supports the tradition but does not prove what the wood was used for. Because of the relic the basilica is known as the Bethlehem of the West. In November 2019 Pope Francis gave a small fragment to the Franciscan Custody of the Holy Land, and it was received in Bethlehem at the start of Advent.
The reliquary stands in the Confessio, the sunken chapel in front of the papal altar that Virginio Vespignani built for Pope Pius IX between 1861 and 1864. The relic is there to turn the heart to the Incarnation: the pilgrim kneels before it to adore the Child it recalls.
Good to know before you go: the Confessio lies below the level of the nave and is reached by steps. The basilica's pages on the relic and on visits give no set times for going down to it, so ask an attendant when you arrive; anyone who cannot manage the steps should ask about seeing the reliquary from the level of the nave.