I don't want to plagiarize, but there is a very good Catholic article on this here.
In summary, the significance of swaddling clothes lies both in the fact that it is humble, common, ordinary, and also that it ties to Solomon - specifically the Wisdom of Solomon 7, 3-6.
7:3 And when I was born, I drew in the common air, and fell upon the
earth, which is of like nature, and the first voice which I uttered
was crying, as all others do. 7:4 I was nursed in swaddling clothes,
and that with cares. 7:5 For there is no king that had any other
beginning of birth. 7:6 For all men have one entrance into life, and
the like going out.
Per the author of the article, this tie is a reinforcement of Christ's place in the Davidic line, and his rightful place as heir to the Throne of David.
This may or may not be agreed upon by all denominations, (The Wisdom of Solomon isn't in the Protestant Bible) or even all Catholics. The idea of it signifying humble beginnings for the King of Kings does seem to be more universal.
Another article at http://living4jesus.net/dynamic/in.swaddling.htm tells that swaddling cloth used in Scripture is not swaddling clothes that a normal infant would be buried in, but rather the strips of cloth used in burial rites.
Have you ever wondered about "swaddling clothes"?
Everyone knows that Jesus was born in a manger, and the 2nd chapter of
the gospel of Luke says that he was wrapped in "swaddling clothes" and
laid in a manger. In fact, even hospitals today use a cloth that they
call a "swaddling" cloth. Unfortunately, it's a misnomer that will
soon be over 2,000 years old, because "swaddling clothes" have nothing
to do with a child being born!
In the Middle East, people traveling long distances were often met
with many hardships and trials on their journeys. In the event of a
death in travel, the body could not continue to be transported for
many days. For that reason, travelers wrapped a thin, gauzelike cloth
around their waist many times. If someone died on the journey, the
others would use this cloth, referred to as "swaddling clothes" to
wrap the corpse in before burying them.
When Jesus was born, there was no room in the Inn, and so Mary and
Joseph used a nearby stable for Jesus' birth. With no other cloth to
use, Jesus was wrapped in Joseph's "swaddling clothes" - the cloth
normally reserved for a person's death.
The King of Kings and Lord of Lords came into this world in a lowly
manger and was wrapped with burial clothes - in truth, He was born to
die. To die for the sins of all mankind.
Clearly, there can be many deeper meanings read into the significance of Swaddling Clothes. Whether all are accurate, or simply a part of our tendency to romanticize and look for deeper meaning, I'm not sure, but I thought that each of these was worth sharing.