The NET Bible translates this as:
In him you also were circumcised—not, however,1 with a circumcision performed by human hands, but by the removal2 of the fleshly body,3 that is, 4 through the circumcision done by Christ.
That's four footnotes in one verse, all related to translation issues:
tn The terms “however” and “but” in this sentence were supplied in order to emphasize the contrast.
tn The articular noun τῇ ἀπεκδύσει (th apekdusei) is a noun which ends in -σις (-sis) and therefore denotes action, i.e., “removal.” Since the head noun is a verbal noun, the following genitive τοῦ σώματος (tou swmatos) is understood as an objective genitive, receiving the action of the head noun.
tn Grk “in the removal of the body of flesh.” The genitive τῆς σαρκός (th" sarko") has been translated as an attributive genitive, “fleshly body.”
tn The second prepositional phrase beginning with ἐν τῇ περιτομῇ (en th peritomh) is parallel to the prepositional phrase ἐν τῇ ἀπεκδύσει (en th apekdusei) and gives a further explanation of it. The words “that is” were supplied to bring out this force in the translation.
(I don't pretend to understand all that as I am a mere novice in Greek (or any other ) grammar.)
As for what Paul means by this, I take it as an allusion to something Moses said:
“And now, Israel, what does the LORD your God require of you, but to fear the LORD your God, to walk in all his ways, to love him, to serve the LORD your God with all your heart and with all your soul, and to keep the commandments and statutes of the LORD, which I am commanding you today for your good? Behold, to the LORD your God belong heaven and the heaven of heavens, the earth with all that is in it. Yet the LORD set his heart in love on your fathers and chose their offspring after them, you above all peoples, as you are this day. Circumcise therefore the foreskin of your heart, and be no longer stubborn. For the LORD your God is God of gods and Lord of lords, the great, the mighty, and the awesome God, who is not partial and takes no bribe. He executes justice for the fatherless and the widow, and loves the sojourner, giving him food and clothing. Love the sojourner, therefore, for you were sojourners in the land of Egypt. You shall fear the LORD your God. You shall serve him and hold fast to him, and by his name you shall swear. He is your praise. He is your God, who has done for you these great and terrifying things that your eyes have seen. Your fathers went down to Egypt seventy persons, and now the LORD your God has made you as numerous as the stars of heaven.—Deuteronomy 10:12-22 (ESV)
It's a theme that Paul returns to over and over again in places like Ephesians 2:11-22 and Romans 2:25-29. Less directly, he talks about needing to die or put aside our natural selves in order to obtain spiritual bodies in passages like Galatians 2:15-21 and 1st Corinthians 15:35-49. I don't think this passage is so much saying something new as saying the same thing in a new way.
Finally and further afield, the idea Paul expresses seems so much like something C. S. Lewis wrote in The Dawn Treader:
"Well, he peeled the beastly stuff right off—just as I thought I'd done it myself the other three times, only they hadn't hurt—and there it was lying on the grass: only ever so much thicker, and darker, and more knobbly-looking than the others had been. And there was I as smooth and soft as a peeled switch and smaller than I had been. There he caught hold of me—I didn't like that much for I was very tender underneath now that I'd no skin on—and threw me into the water. It smarted like anything but only for a moment. After that it became perfectly delicious and as soon as I started swimming and splashing I found that all the pain had gone from my arm. And then I saw why. I'd turned into a boy again. You'd think me simply phony if I told you how I felt about my own arms. I know they're no muscle and are pretty mouldy compared with Caspian's, but I was so glad to see them.
"After a bit the lion took me out and dressed me . . . in new clothes—the same I've got on now, as a matter of fact. And then suddenly I was back here."
After this point in the story Eustace Scrubb becomes a full member of the crew of the Dawn Treader very much as if he were baptized and initiated.
Given Paul's other writings, I think we can understand "body of flesh" here to mean something of the old life that must be miraculous removed in order to commence living a new life as a part of the church.