How is the manger a sign?
Signs were things like earthquakes and comets, floods and droughts. Things outside the control of man. God would use these things to prove to people that a prophet was from Him and had a message from heaven to deliver. If the sign they foretold came to pass, it would mean their prophecy was legitimate.
But a manger? The angels told the shepherds that the sign of the Messiah would be him as a baby lying in a manger. A feeding trough for animals. How is that a sign? And why on earth would the Messiah, God’s Son, be in a manger?
God, the perfect creator, arrayed in unapproachable light, came as humbly as anyone ever has into this world. Then He announced it to more of the most humble people of this world. It starts to beg the question, “what is going on here?”