Forgive me if this is unsatisfactory, but isn't it a lot simpler than these answers are making it seem?
Carnal union (sexul intercourse) is that which consummates and constitutes marriage in this culture. Biblically, theologically, it is when the two "become one flesh" Gen 2:24—it is precisely that in which true marriage consists (not mere betrothal, which pledges you in marriage) .
Consequently, joining oneself to a woman in sexual intercourse you are not married to is akin to marrying one woman, and committing adultery with someone else (becoming one flesh with a woman you are not married to).
Specifically, in context, the Christian Church is said to be married mystically to Christ, and to be one with His Body--it is called the "Body of Christ" (presumably due to this spirutal or mystical 'marital untion').
Therefore, to say that to have sex with a woman ("join to a harlot" 1 Cor 6:16a) is to be one with her in body is to say that they are doing that which is alone reserved for the sacred bond of marriage.
St. Paul is calling intercourse with harlots a kind of adulterous sacrilege. Sacrilege because in doing so one joins Christ's body to an adulterer who sins in her flesh (something St. Paul says is a more grave sin than others 1 Cor 6:18). Adulterous because it is not your wife.
In the bigger picture, he tells us that we do not own our own bodies, but were "bought with a price" (6:20) by Christ, and that joinging with a harlot would be dishonoring our bodies body and expelling the Holy Ghost from your body—whose Temple it ought to be (6:19), whereas you are to glorify God with your body instead (which comitting adultery with any real and legitimate wife, or spiritually, Christ, certaintly does not).