Instead of passing legislation to allow gays the ability to get married, he believes in removing the federal government's regulation on marriage, so anybody under the sun can get married (as long as people were willing to marry you). It's not exactly "pro-gay rights". It's more "pro-liberty".
With one huge caveat: He's stated he supports "traditional" marriage, and his own state's ban on gay marriage. So, it's not quite "get government out of marriage entirely!" It's "get the federal government out of marriage entirely, and let states discriminate against gays."
...So you can see why this stance is repugnant to many LGBTQ people, and others who don't want to live in a society with second class citizenship enshrined in law. Paul's viewpoint isn't not pro-gay rights in the slightest. It's not pro-liberty, either, since it would clearly lead to a significant number of states discriminating against gays. His stance is pro-states' rights, which is just code for "if there's discrimination enshrined in law, DGAF as long as it's done locally."
Q: You say the federal government should stay out of same sex marriage issue and leave it, as it has been traditionally left, to the states. Should the court, therefore, strike down the Defense of Marriage Act?
PAUL: I think it's a really complicated issue. I've always said that the states have a right to decide. I do believe in traditional marriage, Kentucky has decided it, and I don't think the federal government should tell us otherwise. There are states that have decided in the opposite fashion, and I don't think the federal government should tell anybody or any state government how they should decide this. Marriage has been a state issue for hundreds of years. DOMA is complicated, though, because DOMA does provide protection for the states from the federal government. But, then part of it federalizes the issue. I think the way to fix DOMA is maybe to try to make all of our laws more neutral towards the issue, and I don't want the government promoting something I don't believe in.
And also:
Rand Paul said that affording the distinction to marriage to same-sex couples "offends myself and a lot of other people." In an interview with Fox News, the Kentucky Republican, who described himself as a "libertarian conservative," made the remarks when asked about his views on gay rights: "I'm for traditional marriage," Paul said. "I think marriage is between a man and a woman. Ultimately, we could have fixed this a long time ago if we just allowed contracts between adults. We didn't have to call it marriage, which offends myself and a lot of people."
Source is: http://www.ontheissues.org/Domestic/Rand_Paul_Civil_Rights.htm
tl;dr: Paul's happy with gay people being second class citizens. He just thinks they should be degraded on a local, rather than national level. And that's definitely preferable to them having formal equality on a national level.