The Sanctuary of Greccio is built into a rock face above the valley, at about 700 meters (2,300 feet). Its Chapel of the Nativity is the cave of the 1223 celebration and holds a fresco showing two Nativities side by side, Bethlehem and Greccio. Francis was most likely on his way back from Rome, where his Rule had been confirmed on November 29, 1223. Pope Francis signed his letter on the Nativity scene, Admirabile signum, at Greccio on December 1, 2019; it says the caves here reminded the saint of the countryside of Bethlehem, and that he wished to see with his own eyes how the Child lay in a manger, on hay, between an ox and a donkey.
Greccio is one of four small Franciscan sanctuaries that give the Rieti Valley its name of Holy Valley. Fonte Colombo is associated by tradition with the writing of the Rule, and Poggio Bustone with Francis's early preaching with his first companions; La Foresta is the fourth. They sit on different hillsides, so seeing all four takes a dedicated day with a vehicle. Walkers link them on the Cammino di Francesco, a waymarked circuit of some 80 to 90 km (50 to 56 miles, depending on the source) walked over several days, with a pilgrim passport stamped at each sanctuary.
Good to know before you go: Greccio is about 97 km (60 miles) from Rome, a detour off the usual road to Assisi. The sanctuary is reached on foot by paths and stone staircases, and it is a working friary, so go quietly. The opening hours could not be confirmed from the friars' own site for this answer and are not given here.