Bethlehem is important to recognize as Jesus’ birthplace to show that the children of David’s house are no longer born in Jerusalem (the city of David). Instead, the Edomite Herod rules in Jerusalem; however, Jesus’ birth is still announced in Jerusalem. He is both an outsider and the rightful King.
The first thing mentioned in Matthew 2:1–23 is the name of Jesus’ birthplace which stands in clear contrast to Jerusalem, which the evangelist mentions right after Bethlehem. A birth in Jerusalem, which had been the norm for sons born into the family of David prior to the exile, was no longer a given in the post-exilic era. Strangers now rule in the city of David. In Jesus’ days it was Herod, an Edomite king. The fact that he is expressly mentioned in Matthew 2:1 qualifies and characterizes the circumstances of Israel during this time (see Matthew 2:1–17). Yet even though Jesus was born outside of the city of Jerusalem and under a foreign ruler, his birth was announced in this royal city.1
1 Jésus étant né à Bethléhem en Judée, au temps du roi Hérode, voici des mages d'Orient arrivèrent à Jérusalem,