Matthew begins this account of the Wise Men by setting it in an historical context, the reign of Herod the king (Herod the Great, AD 47 BC - 4 AD).  This means that in order to reconcile this event with our calendar, Jesus was born in 4 BC!  
Herod the Great was not popular with the Jews for several reasons.  1) He was not a pure Jew, but an Idumean.  (Idumea lay to the South and South East of Judea and the Dead Sea.  Some associate Idumea with Edom and the Edomites, but this is only part of the Idumean heritage).  2) He was a Roman appointee to Govern Israel.  3) Herod, as we shall learn, was a harsh ruler.  Even the Roman Emperor, Augustus, had acknowledged the ruthlessness of Herod.

It is difficult to identify the wise men with any degree of certainty.  The word used to describe them here is that they are magoi  (magoi magoi could refer to magicians or a priestly caste from Persia.)  In all probability they are a group of Persian priests who who are familiar with astral readings (stars).  When the wise men (not three wise men, since Matthew does not number them1) see the star in the East and follow it they come to Jerusalem and Bethlehem anticipating something important.  they recognize that a new king was born.  We do not know all that this involves, only that it was significant enough to bring them to Jerusalem.  They enquire where the new king is.  Word reaches Herod and he is disturbed.  Herod assembled the chief priests and asks where the Christ is to be born.  That is fascinating!  Herod immediately identifies this with the Messianic expectation of Israel.  So do the chief priests, since they immediately cite the prophecy in Mic 5:2.  Careful examination of Matthew's use of this text demonstrates that Matthew is quoting from neither the Septuagint, nor the standard Massoretic Hebrew text.  Matthew's Midrashic paraphrase use of this text was in keeping with Jewish Rabbinic practice, when a "Rabbi" wanted to make a theological emphasis from a text.  The point here is that Matthew reinforces that this text is a Messianic one, readily recognized by the chief priests!  Herod and the chief priests acknowledge that the Messiah is to be born in Bethlehem.  Herod sends the wise men to Bethlehem, a small town about 5 miles South of Jerusalem.

The wise men (Persian Gentiles) acknowledge Jesus as the king and worship him and give him gifts of costly aromatic herbs or ointment.  Being warned in a dream not to return to Herod they leave for their own home country.

The chief point of this story is the Herod and the chief priests do not receive him and pay him homage.  (Of all people, the chief priests should have welcomed and worshiped the Messiah.  Jn 1:11 records "He came to his own home, and his own people received him not.")   

But Gentiles from afar worship him!  This will be a major theme that Matthew will work into his Gospel as it progresses.

Again we are introduced to Matthew's divine intervention theology, an angel of the Lord warns Joseph to flee into Egypt.
Matthew introduces the second of his formulaic fulfillment prophecies in which he makes a Midrashic paraphrase of an Old Testament text.  In this case he cites Hos 11:1.  Again, Matthew quotes from neither the Septuagint nor the Massoretic text.  The Hos text is actually in regard to the nation of Israel.  Matthew recognizes the Corporate Personality nature of the text and uses it in regard to the Messiah.  His purpose in doing this is to show that Jesus is a significant part of God's saving activity in history (heilsgeschichte).  What is happening to Joseph, Mary, and Jesus is not an accident or last minute thought, it is part of God's ongoing work of salvation in history.

Herod's response is top have all male children in Bethlehem who were 2 years old and under, killed.  Matthew cites Jer 31:15 as a prophetic reference for this.  Again, Matthew is taking an Old Testament text and paraphrasing it.  This text, the third of the formulaic fulfillment texts in Matthew, is from neither the Septuagint nor Massoretic text, but is a Matthean paraphrase.  In Jer 31:15 this text is in the context of God's Restoration of Israel.  Matthew applies it to the present situation of the weeping over the cruel work of Herod, again using this to connect the present happenings with an Old Testament fulfillment and God's heilsgeschichte.

In the next paragraph, we again see God's divine intervention in the life of Jesus.  An angel of the Lord tells Joseph to return to Israel.  Herod is now dead and his son, Archelaus was reigning over Judea.  Archelaus was not a king, but an Ethnarc (a ruler of people, a governor of a district.  After Herod's death no-one was appointed king of Judea.)  Archelaus, like Herod was not a pure Jew, being half Idumean and half Samaritan.  Because of his reputation, Joseph was warned not to return to Judea (Bethlehem and Jerusalem), bit to the district in the North, Galilee, over which Archelaus was not governor.  Joseph and his family settled in Nazareth.  Matthew resorts to the fourth formulaic fulfillment text to support this.  The problem is that this text again is not drawn from the Septuagint.  Nor is it drawn from the Massoretic text.  In fact, it is almost impossible to find its precise source!  This would not be a problem for Jews familiar with Midrashic application of Old Testament passages.  Matthew's purpose is again to connect Jesus with, or place him firmly within God's saving activity in history (heilsgeschichte).  If the background behind this text is the Nazaretic vows of Jud 13:5, 7; and 16:17, then we might see a connection with Jesus' dedication to his divine purpose.