This statement is both an example of the problem, and an explanation of it. You begin by throwing everyone in the same pool, “I don’t fear LGBT people, therefore no one does,” then go on to state that those against homosexual marriage are being labeled as homophobic (I’m inferring you think this is a problem and I agree).
Broad statements that stereotype are so much easier to understand than reality, and that’s why we tend to use them. If I inform everyone here that I’m for gay marriage, you’ve already made several assumptions about me. This is how the brain works, and that’s fine in many situations, but it often leads to misunderstanding in public discourse.
I know many Catholics aren’t very homophobic, if at all. So many Catholics I know have great relationships with people that are gay, but still oppose gay marriage because they aspire to follow the morals given by God rather than their own personal opinions. This is very noble.
Myself, I’ll admit I’m a bit homophobic. It makes me uncomfortable to witness two men or women kissing because I can’t empathize. What I’ve learned through prayer and reflection is that homosexual people are different than me, but should not be treated differently. I’ve found that for me, my joy in life is loving my wife and family. As such, I’d never deny anyone that joy.