|Giovanni Bellini (1430-1516), Circumcision of Jesus|
Today, January 1, is the Solemnity of Mary, the Mother of God, a holy day of obligation in the Catholic Church. But on older calendars, since it falls on the eighth day of Jesus' life (according to traditional Jewish and Roman reckoning), it was — and still is, by traditional and Eastern Catholics, as well as some Lutherans — celebrated as the Feast of the Circumcision.
Now, if you focus on the fact of the circumcision itself, you can get kinda creeped out; just to add to the creepiness, the story goes that at one time there were so many people claiming to possess as a relic the Holy Prepuce that one pope forbade further mention of it on pain of excommunication. (That's the story, at least. The reports and claims of faked relics have themselves suffered from inflation and exaggeration over the years; while forged first- and second-class relics were a problem for a long time, serious scholars of the medieval period doubt they were ever so abundant as has been alleged.)
Part of the problem is semantic, even cultural. "Celebrate" has taken on implications of joyous fun that, when the rite emerged, weren't necessarily present in its Latin roots. The noun celebratio and the verb celebro, celebrare in their primary uses referred simply to the act of gathering or thronging, especially for a ritual, and only secondarily to commemorating a special occasion; the Eucharistic meal as a feast can hardly be called festive. Think of "holiday", and how it began life as a contraction of "holy day": holidays became vacations because holy days entail rest from labor.
The significance of the Lord's brith is that it symbolizes the bonding of Jesus to the covenant of the Law. In his nativity, God becomes human; in his circumcision, God becomes a Jew. From this attachment to a particular communion and community, He can then work to extend salvation outward, from Judea to the farthest corners of the earth.
"Think not that I have come to abolish the law and the prophets; I have come not to abolish them but to fulfill them" (Mt 5:17). To fulfill the Law, Jesus had to be subject to the Law as much as any Jewish person: to be counted among his people, the Governor must himself be governed. Even looking at it from a purely human perspective (Jesus as Jewish reformer), the true Reformer doesn't come from outside, and doesn't go outside, that which he wishes to reform; those who do engage in mere innovation. Moreover, Jesus had to be (to use St. Paul's phrase) "perfect as touching the Law" in order to be the unblemished Paschal sacrifice; to be perfect according to the Law, he had to be accountable to the Law in a way that no Gentile, even God-fearing Gentiles like the centurion Cornelius (vide Ac 10), could approach — he had to not only publicly practice Judaism but be known to be Jewish from birth.
So from the standpoint of soteriology, Jesus' circumcision wasn't simply a rote observance of an ancient and somewhat macabre ritual but was rather a necessary step towards establishing his "street cred" as a Jew and a descendant of David, without which he could not claim the role of Messiah. Just the idea of the Ruler of the Universe being bound by His own laws provides plenty of material for reflection.
The question, then, is no longer "Why do Catholics celebrate that?" Rather, the question is: Why don't other Christian communions celebrate it?