Summary: I have no desire to summarize the meaning of the incarnation in Luke's Gospel. This passage has layers and layers of meaning for us to draw on this year and every year. I offer this as a way to hopefully point toward something in the passage that can help launch your reflection and preaching.
Words I found interesting:οικουμενη(ν) ("world", 2.1) The word for "all the world" here really means civilized world, coming from the Greek work οικος. It is a reminder that for those in the Roman empire, this meant the ENTIRE world.
δογμα (literally dogma, meaning "decree", 2.1) No important theological consideration. Just that Rome has always been interested in promulgating dogma ;-)
απογραφη ("registration" 2.2) A few directions one can go with this word.
First, power of Rome: Liddell Scott refers to this as "a register of persons liable to taxation." Rome wanted a census because they wanted to tax and conscript people.
Second, challenge of history: Quirinius doesn't add up in terms of a chronology. https://en.wikipedia.org/wiki/Census_of_Quirinius. (Other scholars are more generous.)
Third, sin of a census: In 2 Samuel 24:10, David confesses to sinning as he has engaged in a census. Why is this a sin? Because the idea was not to count your troops but to trust the Lord in battle. In fact, it may be that the zealots (mentioned in the New Testament) arose out of anger of this census being taken.
Can we put this altogether: Even if you cannot accept as historical fact the coincidence of Jesus birth with the census, Jesus would have been a young child during a census, a brutal reminder of the power of Rome, a foreign and pagan power. Quirinius' biography is a great story of the "Roman dream" where someone rose through military victory and shifting political allegiances. In short, Luke's setting the stage is correct: The Jews existed under an imperial power, hostile to their faith. Jesus was born in an empire that cared not for him. This imperial power was and remains the envy of all other empires in its military and administrative might.
To drive this point home, the angle proclaims, "who is Christ, the Lord." In Greek, this is spelled Χριστος κυριος, which is the basic confession of faith (Christ is Lord) that ran contrary to the Roman confession of faith (Caesar Kurios). The angle here offers a subversive confession of faith!
καταλυματι ("inn", 2:7) There was no room for them in the inn. Later Jesus will make room for himself in another inn -- the upper room (22:11)
μεγαλην (literally "great", 2:9 and 2:10). Two things are great in this passage -- there is a great fear and then a great joy. This sets up, in many ways, the background for the whole of Luke's Gospel: Jesus will cause great fear, but also great joy. It is a backdrop for any good Christmas sermon too -- there is great fear in our world, but because of Jesus, we have reason for great joy.
ευδοκιας ("pleasure", 2:14) I often wondered about this word -- did God intend peace for all people or just those whom he liked? First, the Greek has a textual problem. The manuscripts seem divided (and even in manuscripts there are edits) whether this should be read as a nominative or genitive.
If we read it as a nominative:
N) glory to God; peace on earth; good will among humans (i.e. three items distributed in three realms)
If we read it as a genitive
G) glory to God; peace on earth among humans of (his) pleasure.
If we go with option N) it seems that good will is toward all people, unambiguously. Unfortunately, the evidence textually, even though divided, favors option G).
So, if we go with option G) we encounter a bit more ambiguity. If this is the case (okay, bad pun there), Luke writes "upon the earth peace among people of pleasure/desire." The Greek leaves out the phrase, "of him." It simply states, "among people of desire." I am not sure if we can, on the basis of grammar, solve this case (again, bad pun). What is unambiguous is that God intends for peace on earth! What is ambiguous grammatically and historically is how we humans live into this peace.
ρημα (literally "herema" meaning "word", 2:15) This word is like logos, and it can mean thing or matter or word. That I want to point out is that the shepherds literally say, "Let us behold the word." John's Gospel is famous for such a construction (John 1:14), Luke has the same concept embedded here.
Ιωσηφ (literally "Joseph", 2:16) Just a reminder that Joseph isn't left out of the picture!
συμβαλλουσα (literally "symballoo", meaning "ponder", 2:19) Mary "pondered these things in her heart." The word for ponder is symbol -- to draw meaning, to pull together or literally to throw together. This is fascinating that Mary is gathering together the images and thoughts of the angels in her mind.
Grammar Review: Cognate Accusative
It is considered poor English to write a sentence in which the verb and object share the same word root. For example: I climbed a climb or I rode a ride. We are trained to make the object and verb different words: "I climbed a mountain" or "I rode a bike."
Because of Hebrew's limited vocabulary as well as the importance of simplifying stories for oral transmission, cognate accusatives are very common. Not so much in Greek, however. Which is strange then that Luke uses two of them in this passage:
φυλασσοντεσ φυλακας (literally "guarded their guard," or "tended their flocks," 2.8)
εφοβηθησαν φοβου (literally "feared a great fear," 2.9)
Not sure why Luke does this other than to speculate he was reading a lot of the Old Testament as he wrote the Christmas narrative!