One man is credited with creating the Christmas crib more than any other, and that is the 13th century Saint Francis of Assissi. In 1220, Francis made the pilgrimage to Bethlehem. While there, he saw how Christmas was celebrated in the town of Jesus' birth and was so impressed that he asked the Pope if he might recreate something like it in his own Italian home of Greccio.
And of course during the Medieval period, mystery plays - with mummers acting out stories from the Bible - were one of the most popular forms of Christmas entertainment. Both survive to this day, but it is normally children who act out the Nativity story itself and not the adults anymore.