The phrase come from the angel's announcement of Jesus' birth to the shepherds in the field. Let's look at in its full context:
"The angel said to them, “Be not afraid; for behold, I bring you good news of a great joy which will come to all the people; 11 for to you is born this day in the city of David a Savior, who is Christ the Lord. 12 And this will be a sign for you: you will find a babe wrapped in swaddling cloths and lying in a manger” (Luke 2:11-12). swaddling clothes will be part of the "sign" to them of his true identity as the Savior (Luke 2:12). Why is that?
To fully understand the importance of Jesus' swaddling clothes we need a little background.
In Luke's Gospel it is abundantly clear that Jesus is the "Messiah," which means the "anointed one", in Greek the christos--he is the Christ, the Son of David. Of course, the Davidic association with Jesus' role is obvious if you know the Old Testament. In recounting his last words, 2 Samuel 23 calls David, "the anointed of the God of Jacob, the sweet psalmist of Israel" (2 Sam 23:1).
Jesus' role as the Messiah, the Christ, the Son of David, is emphasized throughout the beginning of Luke's Gospel. At the annunciation, the angel Gabriel tells Mary that her son
"... will be great, and will be called the Son of the Most High; and the Lord God will give to him the throne of his father David, 33 and he will reign over the house of Jacob for ever; and of his kingdom there will be no end" (Luke 1:32-33).
has raised up a horn of salvation for us in the house of his servant David" (Luke 1:69).
The Davidic themes continue in the story of Jesus' birth. He is born in Bethlehem, the city David had come from (cf. 1 Sam 16). In fact, Luke makes the Davidic association with Bethlehem explicit. "And Joseph also went up from Galilee, from the city of Nazareth, to Judea, to the city of David, which is called Bethlehem, because he was of the house and lineage of David" (Luke 2:4).
As mentioned above, the angels also allude to Jesus' role as the Son of David in their announcement to the shepherds: "for to you is born this day in the city of David a Savior, who is Christ the Lord" (Luke 2:11).
So what's the significance of Jesus' appearance? Why is it a sign that the royal son of David is born in such a humble setting rather than into extravagance?
I would submit that it is because Solomon, the first son of David, was also known for such humility (until, of course, he turned away from the Lord). In Wisdom of Solomon 7, Solomon recounts how asked not for wealth or power, but only Wisdom. The son of David goes on to describe how, even though he is the greatest of all the kings of the earth, he was born like all others:
Jesus' humble birth is a sign that he really is the true Son of David. Like Solomon (before he fell away from the Lord), he is not about power, might or glory. Indeed, it would seem that along these lines the fact that he has even more humble birth might suggest that he is somehow even more worthy of glory. And, of course, he too is wrapped in swaddling clothes.And when I was born, I began to breathe the common air, and fell upon the kindred earth, and my first sound was a cry, like that of all. 4 I was nursed with care in swaddling cloths. 5 For no king has had a different beginning of existence; 6 there is for all mankind one entrance into life, and a common departure. 7 Therefore I prayed, and understanding was given me; I called upon God, and the spirit of wisdom came to me. 8 I preferred her to scepters and thrones, and I accounted wealth as nothing in comparison with her. 9 Neither did I liken to her any priceless gem, because all gold is but a little sand in her sight, and silver will be accounted as clay before her. 10 I loved her more than health and beauty, and I chose to have her rather than light...
The Son of David, the Messiah is born. Glory to God!