The crux of his argument is that rights are a zero-sum game, and that extending a new right to one group necessarily takes rights away from another group.
In the long run, the victory of the homosexual-rights faction would be incompatible with the religious freedom of those who oppose such rights – the freedom, for example, to teach one’s children that homosexuality is wrong and not conducive to ultimate happiness, or to run a private university in which the practice of homosexuality is grounds for dismissing a student or an employee.
Hancock's argument doesn't work if you consider rights only from a legal standpoint. For instance, there are numerous laws in place to protect the rights of minorities, and there have been for decades. But this in no way abridges a parent's right to teach his children that Mexicans are lazy, or that Jews are greedy, or that Mormons have horns and worship the devil. And it hasn't legally prevented churches from continuing various racist policies. The Mormon church continued official discrimination against blacks until 1978. Does Bob Jones University still prohibit interracial marriage? If not, this was only a recent change.
So, from a legal standpoint, adding to the rights of blacks and other minorities did not take away the rights of parents and churches to be as bigotted as they chose.
However, when society as a whole decided that blacks should have the same rights as whites, it became much less comfortable for a bigot to continue to hold and express his racist views.
Just the other day, the following scene played out at a dinner at my parents' house:
SIL: Grandmother, how long have you been staying with Bob and Amy?
GM: I don't STAY with them. I LIVE with them. Black people STAY.
<shocked silence from the entire room>
SIL: Ha, ha. Girls, Grandmother's from another generation. And from the South.
GM <clearly annoyed>: That's right. Grandmother's old and from the South.
My grandmother was quite put out that her racist statement was met with shock and disapproval. Perhaps she even felt that she no longer had the right to express herself as she chose. But it is only social expectation that has changed over the last few decades. Certainly my grandmother had every legal right to say what she did. She just doesn't have the right to say it and have everyone agree and approve.
But she never had that right. No one ever does, or ever has.
Likewise, if gay marriage were allowed--even if the entirety of the nebulous "gay agenda" were passed--people would still have the legal right to teach their children to hate gays. Churches would have the legal right to discriminate against gays.
But there would certainly be increasing social pressure to drop those behaviors. Much like BYU and the church felt increasing pressure to change their policies with regards to blacks in the 1970s, they would undoubtedly feel pressure to change their policies with regards to gays in the decade or so after national gay marriage became legal.
People and private organizations have the right to hold whatever beliefs they choose. But they don't have the right to make society approve of those beliefs. Such a right has never existed.