Christians use Micah 5:2 as a prophesy of Christ.
Matthew and perhaps Luke (1:52?) use Micah’s mention of Bethlehem in their nativity stories. Bethlehem means the house of bread, a fitting place for the one Christians call the Bread of Life to have been born. But Micah was writing 700 and some years before the nativity. I think many Christians think of prophets as people supernaturally empowered to see centuries into the future. Micah, however, probably spoke to concerns closer to his own time.
The gospel writers, though, could not fail to see how appropriate this passage was to what they believed did happen at Bethlehem.
Micah was a village elder from a small town, an advocate for the “people of the land”. So his point about Bethlehem may have been that just as God had brought David from this seemingly obscure place to be the king and savior of Israel, so in the future God would again act in the same way bringing salvation out of a place that was “little among the clans of Judah.”
Jerusalem was the seat of the palace and the Temple because one could defend it against foreign armies. But Bethlehem held memories of the real people who had given the Hebrew people their heritage. Many of them were women.
Rachel, the best-loved wife of Jacob, had died giving birth to Benjamin and been buried nearby. Micah refers to this when he uses the old name Ephrathah echoing Genesis 35:19 (see also Ruth 4:11). She was the mother of Joseph and Benjamin. Perhaps people from those tribes made pilgrimages to her shrine in Judah.
Naomi was from Bethlehem and her Moabite daughter-in-law, Ruth, came there. She married Bethlehem native, Boaz. The gospel genealogies name Boaz and Ruth as ancestors of Jesus. This introduces a hint of a more extensive salvation including non-Hebrews, like Ruth.
Once you read beyond verse 2, you see that Micah was predicting a political and military salvation. He had in mind the Assyrian crisis of his own day. You would have to look to the Maccabees or the reestablishment of the Jewish state after World War II for a literal fulfillment, though the literal coronation of a Davidic king happened in neither case.
So at Christmas Christians do not insist that this passage was literally fulfilled in Christ. The New Testament’s use of the prophets is pretty loose. The most you can say is that the reference to Bethlehem is suggestive of the Christmas story.
Some would claim that the early Christians made up the Christmas story based on such suggestive Hebrew Bible passages. Jesus was later known as Jesus of Nazareth, so historical Jesus scholars have said he must have been born there. It would not destroy my faith if that were the case. The Easter story, not the Christmas story, is bedrock.
However, I note that even though Matthew and Luke have nativity stories that do not really fit together (in spite of what we do by putting the Magi and the shepherds together at the manger in Christmas programs and nativity scenes), one of the very few things both accounts agree about is that Jesus was born at Bethlehem.
The simplest explanation for this common tradition is that it happened.
Luke has an explanation about why they were there–the decree from Caesar Augustus. Historically, this is hard to verify. But Micah’s explanation is spiritual. The one who comes forth from Bethlehem will be for God. This introduces the presence, guidance and providence of God. Jesus was born in this “little” place because God willed to begin great events in a seemingly insignificant place.
The character of God is such that the places and people the world sees as great and famous get bypassed. Luke and Acts tell how the gospel moved from Jerusalem to Rome. But it didn’t start in either of these great cities. It started in a dinky little town in the countryside, the town of Rachel, Naomi, Ruth, and ultimately of Mary.