How did the priests find out where the King of Israel was to be born?
The priests did not know initially; when questioned, they searched the Old Testament Scriptures and found in Micah that the promised King would be born in Bethlehem. They had to dig them up, because they had refused to search the Scriptures beforehand.
Les explains that when Jesus presented Himself as the King of Israel, Israel should have known from the Old Testament, but they were unbelieving and refused to search the Scriptures. The wise men came to Oherod asking where the one born King of the Jews was, which frightened him. He then went to the priests and asked where the King of Israel was to be born. They did not know offhand, so they dug up the Old Testament Scriptures and found in Micah that He would be born in Bethlehem. Les notes they should have known immediately and told Oherod, along with the other things attendant with His coming, but they did not. The same pattern appears in Romans 11: Israel should have known but refused, remaining in unbelief.
But thou, Bethlehem Ephratah, though thou be little among the thousands of Judah, yet out of thee shall he come forth unto me that is to be ruler in Israel; whose goings forth have been from of old, from everlasting.