If anybody does not think that Matthew 1 presented Jesus as king, then they might have a bit of a tough time understanding why the wise men come to Jerusalem asking "Where is the child who has been born king of the Jews?" (v. 2)
Now, there are two ways (at least) we can understand the literary effect of including the wise men in the narrative. First, we could view them as being St Matthew's way of introducing Herod and the role he plays. Though a valid understanding, since these men do not have any kind of role after chapter 2, I contend that this gives us a very shallow view. The second seems more likely:
Matthew 1 is about King Jesus, finally come, the true son of David, here to save his people from their sins. Now, even in this chapter, we begin to see the role the book of Isaiah is going to play in how Jesus understands and explains his ministry. Since St Matthew gives no reference to Isaiah other than saying "the prophet", I think it fit to infer that the audience in question would have had a decent grasp of the Old Testament (although he makes reference to Jeremiah explicitly in 2:17 before quoting him). I believe St Matthew has passages such as Isaiah 60 in mind, or perhaps Isaiah 49, where the message of salvation finally goes out to the nations. The wise men here represent the first of these peoples who will flock to Jerusalem, flock to the holy mount Zion and hear the word of God. Right now, these men pay homage to the king of the Jews...but why? I think St Matthew is foreshadowing here the inclusion of the gentiles. Perhaps he is even saying "look, even the pagans pay homage!" before he shows us how the Jews themselves will react to Jesus and his teachings. In that sense, he foreshadows also the Jewish rejection of Jesus as the Messiah.
Herod also appears only in this section in St Matthew's gospel, and his reaction to the wise men is interesting: verse 3 reads "When king Herod heard this he was frightened and all Jerusalem with him." Herod is scared because he knows very well that the Jews are awaiting a king from the line of David, and he can be no such king. Herod has a sort of paranoia that some more active Israelite will plunge a knife in his back or slip something in his drink - so clearly, he is not happy when somebody comes to ask for the king of the Jews. This needs to be dealt with.
Before getting to how Herod deals with this, however, I should comment on the second half of the verse - why is Jerusalem frightened?An anachronistic answer might be that they fear the Messiah will rebuke them, and so they fear his coming. I think it far more likely that Jerusalem is more scared of what Herod will do with this information. Indeed, it is not pretty. Herod tells the wise men to inform him of Jesus' location, and deviously plans to kill him. This is not strictly stated in the text, but Herod is not being very open about this ("Herod secretly called for the wise men..."), and also it is reasonable to infer from what Herod does when this first plan is thwarted - more on that in a moment.
The wise men come to the child Jesus and bring him gifts of gold, frankincense and myrrh from their treasure chests, all with joy. I contend, although there is much room for disagreement, that these gifts are meant to be reminiscent of Solomon, the man with the most gold, and frankincense/myrrh only appear together, to the best of my knowledge, in the Songs of Solomon (see 3:6, 4:6, 4:14). This interpretation fits well with St Matthew's royal portrayal of Jesus, and how the wise men treat him as king, yet the reference may well be obscure. Others have suggested that gold symbolizes his kingship, whereas frankincense his divinity, and myrrh his Passion and death. as myrrh was sometimes used with as a burial ointment (see John 19:39). This was the interpretation of St Irenaeus, and his allegorical interpretation is sound doctrinally, so I have no problem with it. 
The most memorable thing that Herod does within the Bible is the massacre of the innocents, but because of no extra-biblical evidence for such a horrific act, some scholars have suggested that it never happened. From a literary standpoint, it is not senseless to have this story here, since one of the the images that St Matthew takes from the Old Testament and applies to Jesus is Jesus as the New Moses (which we shall get to in particular with the sermon on the mount, which reminisces of Mount Sinai). Moses also lives in a time where the authorities are killing children, and Moses also escapes (this is narrated in the book of Exodus). Although more historical evidence would be preferable to properly establish its occurrence, we must understand that Bethlehem is not a very large village, and so the number of children killed would most likely be very small, and it is completely possible that such an event was not newsworthy to later historians, through whom we have no record of it. Herod's evildoings were numerous enough and much more notable in other instances, that this particular one may well have passed under the radar.
One final comment should be made on typology in this passage, because the Isaiah passage earlier and the Hosea passage of verse 15 both seem to be taken wildly out of context if we suppose that St Matthew is using them as proof-texts, or prophecies yet to be fulfilled that come to fruition in Jesus. The verse from Isaiah, in context, seems to be referring to king Hezekiah, who does appear to rescue Israel from various evils (see 2 Kings 18, in particular verses 1-6). The Hosea passage, in turn, refers to the already accomplished (even by the time of Hosea) calling of Israel (God's first born son, Exodus 4:22) out of Egypt. How then, does Jesus fulfil these finished prophecies? The answer is typology. King Hezekiah indeed rescued Israel from various evils, and King Jesus does all the more. Hezekiah is "God With Us", and Jesus, so much more! Jesus is therefore a "type of King Hezekiah", but an ultimate type - ie, the fulfilment of that prophecy. Hosea is not even being prophetic in the quoted verse, and yet just as Israel was called out of Egypt, so too the true and eternal first-born son of God, Jesus, is called out of Egypt. We should not fall into the trap of thinking that Jesus makes them come true in the sense that they were still open-ended - nor must we forget that Jesus really is the king that rescues us from evil, that he is "God with us" and that he is the Son of God.
 At some point in church history distinctions began to be made between different levels on which the text spoke, and each level gave rise to a way of understanding it. One of these was the allegorical, which does not mean that St Irenaeus was a liberal theologian, simply that he was not reading it as simple history, but also as a literary work. The distinctions were not entirely exact, but in broad strokes some medieval theologians would have a four level system: the literal sense, the allegorical sense, anagogical sense and the moral sense.