The argument is simple. Marriage has traditionally been seen as a union of one man and one woman. Once we decide as a society that the gender of the people in the union no longer matters we will have no grounds for resisting the further step of concluding that neither does the number of people in the union matter. In other words, marriage will inevitably come to be almost anything anyone wants it to be and when marriage is anything at all then it will pretty much cease to exist as anything more than a remnant of what it once was.
The argument is not without its critics. Some have derided it as a mere slippery slope argument, which is a silly objection since there's nothing wrong with slippery slope arguments. Others say that unions of more than two people are just gross, and no one would really want to do it, as if there were not a sizeable number of people who wouldn't do whatever they were legally allowed to do. I'm sure that there are a lot of people who would love even to marry their pets were the law to allow them.
As for marriages involving multiple partners (polyamory) the movement for this particular proclivity already exists and its votaries are biding until gay marriage gains full legal acceptance. When that day arrives polyamorists will begin pushing for their own particular notion of wedded bliss, and what compelling reason could anyone have for denying arrangements in which multiple men enter into matrimonial union with multiple women once we've decided that the gender of the spouses in a marriage no longer matters?
If you doubt that this is the inevitable next step in the marriage debate then read a pair of articles by the pseudonymous polyamorist Michael Carey at Slate here and here. What he says makes pretty clear what challenges will be facing our courts of law in the near future.
Although it's not the point of this post, I'd like to comment on something Mr. Carey says at the outset of the first article. He writes:
In the course of defending their right to treat gay people as second-class citizens, conservatives have frequently deployed slippery-slope arguments: “If we accept same-sex relationships, what will we have to tolerate next? Bestiality? Pederasty? Polygamy?” While these arguments are stupid, the people making them are not, or at least not always. They’re doing their best to trot out a parade of horribles that will shock the sensibilities of most Americans.In other words, Mr. Carey maintains, gay marriage, plural marriage, and traditional marriage are all based on consent whereas those other "horribles" are not. I wonder, though, how long consent would matter in a society that has lost any objective grounds for moral discrimination.
Clearly we should be shocked by violations of consent. (Reminder: Children and animals can’t consent!)
A society that has no transcendent basis for objective moral duties, a foundation that only theism can provide, will eventually subjectivize morality, and when morality is subjectivized it will inevitably devolve into the simple philosophy of might-makes-right. Those who possess the power to do what they wish will find their behavior sanctioned by a society that lacks the moral resources to disapprove any kind of behavior and which will almost certainly lack the will to pass judgment on the behavior of the powerful.
When we reach that point, which may not be as far off as we might hope, the law will eventually be made to conform to the moral consensus, those "horribles" will no longer seem so horrible, and consent will be treated as a quaint anachronism.