The gospel according to St Matthew is one of the two gospels with an infancy narrative, and the Pope Emeritus published some (reportedly) excellent scholarship on it. Since I have not read it and I doubt I could top it, my reflections on this passage will be mostly things that stick out to me, and bits of background information I found illuminating.
"Now the birth of Jesus the Messiah took place in this way."
It is common for very important people (kings and emperors, in particular) in the ancient world to have miraculous birth narratives told of them, and so St Matthew tells of a miraculous birth at the onset of his gospel. From a literary perspective, I think the goal is to elevate this figure Jesus, whom he has already called the Messiah/Christ thrice, to this status of leader, of King. Just because something is a literary device, however, does not mean Jesus was not born of a virgin - indeed, the reference in verse 23 seems to indicate that St Matthew believes this to be an actual event. I personally embrace wholeheartedly the idea that Jesus was born to Mary, the Mother of God - but this is something taken from a richer theological framework, from a broader theology of Scripture and revelation. Nonetheless, that is what the text says: that Jesus, the Messiah, was born of the Virgin Mary.
What about the role of St Joseph? He appears as a rather quiet figure. He is spoken to, but he does not say anything. Similarly, the Church has regarded St Joseph as the quiet father figure, giving him a certain nobility and humility of character. In support of this, the text refers to him as "a righteous man" (v. 19).
Other than his title, the Christ, and his genealogy (being the son of David), we do not know much of Jesus until verse 20. Here we learn that his origin, though Davidic, is also divine: "the child conceived in her is of the Holy Spirit." We also learn in the same verse of his mission, or at least, one of his objectives: "he will save his people from their sins."
When we, as Christians, read this "mission statement" given to Jesus by the angel, we might suddenly envisage the cross, and if we reflect on that image in the context of the annunciation, we may be lead to thinking this is a sad passage; the angel announces Jesus' death even before he is born. This image, though accurate, is not the message I think was trying to be made. Instead, I believe we should try and see this section as the birth of the child of the covenant, the promised son of David who would bring to fulfilment God's plan of salvation that had been begun with Abraham. Here is the person who would set things straight in God's plan, dealing decisively with injustice and evil-doings - that is, putting an end to sin. Notice the wording of the text is not "pay the price for people's sins" or "he will be a propitiation for their sins", but a message of salvation. For the moment, St Matthew is feeding our excitement at how the child Jesus is affirming Messianic expectations - it shan't be long before they are subverted, but not quite yet.