The Magi came to Bethlehem because they saw a sign in the sky. It was a comet which broke into pieces, forming a pattern by the arrangement of the comet pieces and their relation to the stars and planets. They interpreted this, by the grace of God, as the birth of a great king, a king so greatly favored by God that his birth would be written into the stars and the sky.
Blessed A.C. Emmerich describes the Magi seeing the comet break apart: “They were on a pyramid-shaped tower looking through long tubes at the Star of Jacob, which had a tail. The star split asunder before their eyes, and I saw a great shining virgin appear therein, before whom a radiant child hovered in the air.”
(Ancient astronomers in China used to view the sky through long tubes, so that they could see the stars even during the day. It is possible that people in other lands also used this technique.)
The Roman also observed this comet. Cassius Dio describes this comet, saying it “dissolved into flashes resembling torches.”
However, this does not explain how the Magi found the Christ-Child. I believe that, as the Magi approached the holy land, a supernatural (or preternatural) light appeared to guide them specifically to Bethlehem, to the place of the birth of the Savior.