Okay, in this post I'm going to summarize what I found in my commentary — Joseph Fitzmyer's commentary on Luke for the Anchor Bible series.
Fitzmyer seems to share some of my confusion about this text and what he describes as its "flamboyant and grotesque details." However, he lifts up an idea from Bultmann: "The story has made use of the literary motif of the 'duped devil'. The demons sought to control Jesus by pronouncing his name; when asked for their own name, the demons reply not with their name, but with their number. . . When commanded by Jesus to come out of the man, they request to be sent into the pigs instead, rather than into the abyss. The exorcist consents, sends them, indeed, into the pigs, only to make the latter stampede to their and the demons' destruction. Thus Jesus duped the demons and 'saved' the man."
On this reading, the most important thing is Jesus' power over the demons in their struggle. The demons try to exercise control over Jesus by using his name (names carry great power), and suggest a military struggle with the term "Legion." However, Jesus has the final say in the conflict, and the demons are defeated while the man is saved.
Fitzmyer concludes, "The story depicts Jesus using his power to heal an unfortunate demented human being, an outcast of society, thus restoring him to soundness of mind and wholeness of life. This salvific concern is manifested, moreover, even toward one who is presumably a pagan." He also lifts up the importance of word in this narrative: the demons try to gain power over Jesus through his name, Jesus defeats them through word, and the man who was saved spreads the word of Jesus' actions. "He who cured by his word thus becomes the one proclaimed."
This sounds like a good direction to take in my sermon. I don't want to get lost in the details of the pigs, the demons, and the swineherders. This story speaks about Jesus' saving power and about the power of the word being proclaimed by many people in many places.