Well, they believe that being gay is a sin. They believe that it is morally wrong, harmful, and the world would be better off without people being gay.
Why do they believe this? Reasons! I didn't actually give a reason you say? Shut up, that's why!
Oh, and it's in the Bible. Except, it's actually not really. And the place where it in theory is is right next to the thing that says that wearing clothes make of cotton AND polyester is punishable by death. But someones pastor said gayness is a sin, so no actual personal faith or religious introspection is required. Just blind dogma.
Why the blind Dogma? I mean, the only reason for it is the desire to believe in a fair and just universe with a loving God who ostensibly inspires kindness and love and acceptance and...oh. Right. So they believe in a God for, if not an accurate reason, an understandable and moral reason. And once you get caught up in the idea of all powerful, universally loving force of absolute good, you kinda don't notice when some dick sneaks in some anti-gay propaganda into the rules.
Once you arrive at this conclusion, you run into the, "Just World" problem. You believe in a loving God. You struggle with the idea of evil in a world with a loving God. Philosophers have been doing that for centuries, so are uptight moms, if perhaps with a little less sophistication. They have been told that Homosexuality is a sin, and accepted it because everyone takes shortcuts when it comes to your moral code, and that man who said the gay is evil also organized that charity drive to raise money for my friends son's lung cancer treatment or something. So you think gayness is a sin and you run into a problem: If being gay is a sin, how could God make people be BORN gay? That's just a dick move, and God may be weird and distant and completely non-understandable in a normal sense, but he is not a dick. The only explanation is that he didn't, people choose to be gay of their own free will. This means that it is possible for a person to become gay through outside influences. Since this belief is a total fantasy, they have to blame it on society, those people who see that there's no reason to have a problem with gays. Bam, instant alarmism, welcome to crazy town.
Homophobes believe that DC is trying to indoctrinate kids into a gay style for morally centered, understandable (if fallacious) reasons. Doesn't make them any less of a bunch of blithering idiots, but it does explain how they got their.