1. Today scholars suggest that this section of Matthew is not historical. However, the story should be seen as like a parable that is rich in meaning.
2. The star can be seen as a light shining in darkness. The darkness is the ignorance and sin that filled the minds of people before Christ. Although the Jews had more information about God from the law and the prophets than did the gentiles, they erred by not recognizing that God had more to reveal. They were comfortable in their ideas; they thought they had God "in a box" and wished to keep him there. The Magi represent the non-jews who had enough knowledge and wisdom to realize that God was breaking in on history and they responded by seeking out what God was doing. They read the signs and responded.
3. It is worth noting that God reaches out to people through the star to announce his coming, then sending his son. Our response should be that of the magi who then see the signs and respond by searching out the son. We too our called to see the signs that God gives announcing his presence in our lives and world, then search him out ourselves by seeking the son, his teaching, his life and message, then by doing homage to him ourselves through faith.
4. Tradition has it that three kings or wise men can in search of the son. The passage however, does not give a number, and does not say kings or rulers. The magi are wise in the sense they have arcane knowledge, perhaps philosophy or astrology. (Recall that the modern science of astronomy did not exist then.) Because these are people who watch the heavens and who recognize the significance of the star they must have knowledge of astrology, or be in contact with those who did.
5. The Magi probably came from the east which would be present day Iraq and Iran, which in ancient times would be Persia. This would entail a long difficult journey with considerable expense. They would need animals, and provisions. They would have to contend with bandits and uncertain conditions. This should show how much effort we should be willing to expend to search out the Son of God.
6. The key to the parable is the reaction of King Herod and his advisors. He and they along with all Jerusalem were troubled. This might be expected of Herod who was a vicious, paranoid ruler. Josephus reports that he was in constant fear of being killed or deposed and so would go to any length to protect himself. He even wanted to kill a large number of people after his death, according to his will so that "his grave might not be without the tribute of tears. This barbarous command was not carried into effect..."
The "troubled" reaction of Jerusalem however, is more of a puzzle. One would have thought that a sign that God is about to intervene, that He has sent the Messiah should have given them hope and provoked a joyful response. The story indicates this was not their reaction and one must assume that means they feared change more than the status quo of Roman domination under he tyrant Herod. At the very least it shows a lack of trust in God that is a requirement of faith. However, the Jews had a long history of not listening to God's messengers until long after their message.
7. This dichotomy between the Jewish reaction and that of the gentiles represented by the Magi is a prediction of what would happen in the early church. Acceptance of Christ by Jews was limited, while the movement to faith by gentiles was much greater. Eventually, the Jewish component of Christianity would die out leaving only the non-Jewish people to embrace and follow Christ.
8. When the Magi do find the Savior, they give homage. A physical expression of this would be to bow, genuflect or even prostrate oneself. (In Asian cultures this would be a "kowtow" where the forehead touches the ground.) This is an outward sign of relative status. However, it may also signify a surrender of ideas and belief. The Magi would be offering more than their gifts, but the gifts of themselves and their thinking. The old pagan way of life is surrendered to the new life of Christ.