He framed his reaction in terms of equality of opportunity: ""It allowed me to think of this issue from a new perspective, and that's of a Dad who loves his son a lot and wants him to have the same opportunities that his brother and sister would have."
The story is similar to how one of the legislators who blocked a bill that would legalize abortion in Indiana later came to my grandfather (a pediatrician), asking for a referral to an illegal abortion for his 16 year old pregnant daughter. Naturally this politician wanted to ensure that his daughter had the opportunities to succeed, in spite of any mistakes she may have made.
Same-sex marriage is a wonderful example of where John Rawls' concept of the veil of ignorance applies: we don't know whether our children will want to marry someone of the opposite sex or same sex, but we all want them to have the opportunity to marry.
About 3-5% of children of conservative politicians are likely to want to marry the same sex --- the same as in the general population --- so we may continue to see evolution like this on same-sex marriage.
It's too bad that conservative politicians are unlikely to have the same experiences for social policy. Now that social mobility in the US is at a stand-still, the children born in the top quintile of income are likely to end up there themselves, and likewise for the bottom quintile. Few policy makers make decisions informed by the personal experiences of people who are disadvantaged by the current system. They are unlikely to know adults trying to support a household on minimum wage, and may truly believe that health and economic disparities are due to personal failings.
About 3-5% of the US population is gay, and aside from some predictors (e.g., having older brothers), gayness is more-or-less randomly assigned. Not even 3% of children makers are randomly assigned to be socially disadvantaged. If they were, we would have a situation paralleling Rawls's veil of ignorance, and we might have fewer social disparities.