How did they know Bethlehem?

 ⁴ Gathering together all the chief priests and scribes of the people, he inquired of them where the Messiah was to be born. ⁵ They said to him, “In Bethlehem of Judea; for this is what has been written by the prophet:

⁶ ‘And you, Bethlehem, land of Judah, Are by no means least among the leaders of Judah;
For out of you shall come forth a Ruler Who will shepherd My people Israel.’” (Matthew 2:4-6)

This is a reference of Micah 5:2-4, but where did Micah get the revelation from? I believe the Holy Spirit led him to Israel’s prophecy,

¹⁰ “The scepter shall not depart from Judah, Nor the ruler’s staff from between his feet,
Until Shiloh comes, And to him shall be the obedience of the peoples.
¹¹ “He ties his foal to the vine, And his donkey’s colt to the choice vine;
He washes his garments in wine, And his robes in the blood of grapes. (Genesis 49:10-11)

God’s covenant with David

¹² When your days are complete and you lie down with your fathers, I will raise up your descendant after you, who will come forth from you, and I will establish his kingdom. ¹³ He shall build a house for My name, and I will establish the throne of his kingdom forever. (2 Samuel 7:12-13)

And the origins of David being given in Ruth,

¹¹ All the people who were in the court, and the elders, said, “We are witnesses. May the LORD make the woman who is coming into your home like Rachel and Leah, both of whom built the house of Israel; and may you achieve wealth in Ephrathah and become famous in Bethlehem. (Ruth 4:11)

And later,

¹⁷ The neighbor women gave him a name, saying, “A son has been born to Naomi!” So they named him Obed. He is the father of Jesse, the father of David. (Ruth 4:17)

References

Matthew 2:4-6

Micah 5:2-4

2 Peter 1:21

Genesis 49:10-11

2 Samuel 7:12-13

Ruth 4:11, 17 Exodus 25:4