The cave interpretation has some verisimilitude (since people in the Holy Land did use caves for houses or shelter or garages back then), but I think the interpretation really caught on more because of a prophecy in Isaiah 33:16, which in the Greek Septuagint reads: “he will dwell in the high cave of a strong rock.” And this pretty much explains why the cave tradition disappeared in the West: not only did Westerners tend to associate stables with specially-built structures rather than caves, the Hebrew version of Isaiah 33:16 (and St. Jerome’s translation of the Hebrew) does not mention a cave - “the fortifications of rocks shall be his highness” - so there was really no prophecy to fulfill.
P.S. I’m one of those who doubt whether the Church of the Nativity was built on the exact spot, though it may be on the right location (Bethlehem). For one, the early tradition (the Protoevangelium of James and St. Justin Martyr) has it that the cave was supposedly at the outskirts of Bethlehem (as Justin says, ‘nearby’), but the current cave - which once also served as a center of the Tammuz-Adonis cult during the 2nd-3rd centuries, at least according to St. Jerome - is apparently well
inside Bethlehem (as Origen says, ‘in’). (Though granted, Justin’s geography was really vague; not to mention that Origen did live in Caesarea - Roman capital of Palestine - for some time.) Really, I think the only sites in the Holy Land associated with Jesus that have a relatively high degree of being the ‘correct’ spots are St. Peter’s house in Capernaum and the Church of the Holy Sepulchre (Golgotha and Jesus’ tomb) in Jerusalem.
From a past post:
Back then, people usually kept their livestock inside the house to protect them from the weather and from theft. Freestanding barns and stables were very rare. It was very practical too: in the days before central heating, the animals’ body heat helped keep the house warm. Not to mention that if you kept a sheep or a cow, you’d have a ready supply of milk and dung (fuel source). One possible way of keeping the animals was by dividing the house into two storeys: the animals would be at the lower level, while the family’s living quarters would be upstairs. Yes, kind of like a garage, a basement, or a back room.
http://willcookson.files.wordpress.com/2011/12/palestinian-house-one.jpg?w=440
http://willcookson.files.wordpress.com/2011/12/palestinian-house-two.jpg
http://4.bp.blogspot.com/-9ohzPgETfxY/TuZiHJ9iGJI/AAAAAAAAAvw/VQ9Dx0DgdZg/s1600/inn+pic+1.jpg
http://www.bible-archaeology.info/Amman4roomhouse.jpg
http://www.welcometohosanna.com/LIFE_OF_JESUS/JESUS_LIFE_PIX/NAZARETH_VILLAGE_HOME_INT.jpg