Signs in tradition and scripture, can have two basic senses. In the first sense, they are often miraculous indicators of the speaker's trustworthiness, which encourage the listener to believe. In the second sense, they embody or represent a larger or more transcendent truth. The general formula for signs in my first sense is that the speaker first delivers a message, then performs a sign to reassure the listeners of his authority. We can see this by considering a couple other signs in scripture that follow the format.
First, let's consider the interaction between Moses and God as God prepares Moses to speak with Pharaoh for the first time.
1 “But,” objected Moses, “suppose they do not believe me or
listen to me? For they may say, ‘The LORD did not appear to you.’”
2 The LORD said to him: What is in your hand? “A staff,” he
answered. 3 God said: Throw it on the ground. So he threw
it on the ground and it became a snake,a and Moses backed away from
it. 4 Then the LORD said to Moses: Now stretch out your
hand and take hold of its tail. So he stretched out his hand and took
hold of it, and it became a staff in his hand. 5 That is so
they will believe that the LORD, the God of their ancestors, the God
of Abraham, the God of Isaac, and the God of Jacob, did appear to you.
6 Again the LORD said to him: Put your hand into the fold
of your garment. So he put his hand into the fold of his garment, and
when he drew it out, there was his hand covered with scales, like
snowflakes. 7 Then God said: Put your hand back into the
fold of your garment. So he put his hand back into the fold of his
garment, and when he drew it out, there it was again like his own
flesh. 8 If they do not believe you or pay attention to the
message of the first sign, they should believe the message of the
second sign. 9 And if they do not believe even these two
signs and do not listen to you, take some water from the Nile and pour
it on the dry land. The water you take from the Nile will become blood
on the dry land. (Exodus 4:1-8)
Now, smarter or more patient men than me can likely read some meaning into both signs. But, on the surface of it, Moses is given several miracles he can perform in front of Pharaoh to attest to his authority, or to the fact that he has been sent by God with his message. And, unless my memory is foggy, Moses appears before Pharaoh, first with his message, and then displays the signs of his authority and mission.
Then, let's consider Jesus' first "sign" according to John.
1 On the third day there was a wedding in Cana in
Galilee, and the mother of Jesus was there. 2 Jesus and his
disciples were also invited to the wedding. 3 When the wine
ran short, the mother of Jesus said to him, “They have no wine.”
4 [And] Jesus said to her, “Woman, how does your concern
affect me? My hour has not yet come.” 5 His mother said to
the servers, “Do whatever he tells you.” 6 Now there were
six stone water jars there for Jewish ceremonial washings, each
holding twenty to thirty gallons. 7 Jesus told them, “Fill
the jars with water.” So they filled them to the brim. 8
Then he told them, “Draw some out now and take it to the headwaiter.”
So they took it. 9 And when the headwaiter tasted the water
that had become wine, without knowing where it came from (although the
servers who had drawn the water knew), the headwaiter called the
bridegroom 10 and said to him, “Everyone serves good wine
first, and then when people have drunk freely, an inferior one; but
you have kept the good wine until now.” 11 Jesus did this
as the beginning of his signs in Cana in Galilee and so revealed his
glory, and his disciples began to believe in him. 12 After
this, he and his mother, [his] brothers, and his disciples went down
to Capernaum and stayed there only a few days. (John 2:1-12)
Admittedly, I feel like I'm contorting John's first sign a little to fit the format, but if notice the portions I've emphasized, the message is clear: do whatever he tells you. If His mother would not have given the message, "do whatever he tells you," the miracle would have been neat, but it would not have been a sign. It would have been a magic trick. And without the sign, the message lacks authority. The wedding guests would have no reason to heed Mary's command to submit fully to Jesus.
This particular sign is also packed full with pretty obvious symbolism, meaning, and power. And in those respects, the sign is a sign in other senses. The miracle extends a wedding celebration, likely indicating that Jesus' mission is about completing our joy (something He explicitly tells us later). It is significant that the jars used are ceremonial jars. Jesus commands that the ceremonial jars be filled with ordinary water, and by His miracle, he reveals them to be significant or holy -- he condones and makes holy the Jewish ceremony.
Now, I don't think the context of your initial quote needs to be broadened, but let's widen the context a little anyway:
8 Now there were shepherds in that region living in the fields and
keeping the night watch over their flock. 9 The angel of
the Lord appeared to them and the glory of the Lord shone around them,
and they were struck with great fear. 10 The angel said to
them, “Do not be afraid; for behold, I proclaim to you good news of
great joy that will be for all the people. 11 For today in
the city of David a savior has been born for you who is Messiah and
Lord. 12 And this will be a sign for you: you will find an
infant wrapped in swaddling clothes and lying in a manger.”
13 And suddenly there was a multitude of the heavenly host
with the angel, praising God and saying: 14 “Glory to God
in the highest and on earth peace to those on whom his favor rests.” (Luke 2:1-14)
First, the angels deliver a message. (That is their purpose, after all!) The message is that the savior and Messiah has been born. And as I understand it, the message above suggests that this savior is both God and man, Messiah and Lord. And then, to lend credence to their message, they basically tell the shepherds, "Go and see for yourself! This is probably the first and last time you'll see a baby in a food trough!"
Admittedly, I don't know how often Jews laid their infants in food troughs. But, these angels make the pretty incredible claim that the Messiah has not only been born, but is God Himself. And as a sign that they're telling the truth, they tell the shepherds where to find Him.
Now, as a Catholic, there's a great deal of additional sign and symbolism here. God is born in Bethlehem (the house of bread), wrapped in swaddling clothes (He is fully human), and has been laid in a food trough, a sign that He has been offered up for consumption -- and not just any consumption, and not even implying reverent consumption, but animalistic, carnal, beastly consumption.
So, the sign of the infant, I think, hits on both senses of a sign. It testifies to the accuracy of the angel's message. And it signals Jesus whole mission: to "condescend" into humanity, thereby making humanity holy, and to submissively offer Himself as the fundamental food for our immortal bodies as regular bread would have been for our mortal bodies. I.e., Jesus' first "act" is to signal (be a sign of) the Eucharist. And we are meant to devour Him, as hungry beasts from a food trough.