How did the wise men know what to look for?

Daniel was given multiple visions that were documented. When a vision was not understood it was stated.

²⁷ Then I, Daniel, was exhausted and sick for days. Then I got up again and carried on the king’s business; but I was astounded at the vision, and there was none to explain it. (Daniel 8:27)

 ⁸ As for me, I heard but could not understand; so I said, “My lord, what will be the outcome of these events?” (Daniel 12:8)

But at the end of the vision that prophesied the coming of the Messiah in Daniel 9, no such statement. I believe he understood; and overseeing all the wise men of Babylon he shared it with them.

³⁵ Yet wisdom is vindicated by all her children.” (Luke 7:35)

¹Now after Jesus was born in Bethlehem of Judea in the days of Herod the king, magi from the east arrived in Jerusalem, saying, ² “Where is He who has been born King of the Jews? For we saw His star in the east and have come to worship Him.” (Matthew 2:1-2)

References

Daniel 8:27

Daniel 12:8

Daniel 2:2, 48

Luke 7:35

Matthew 2:1-2