I have an idea...
I think that because a lot of people who say "I don't agree" also say "I hate..." or "You're nasty, demonic, hell-bound, etc." many homosexuals will equate that with hatred. This, I think, is especially true of those who don't know the Lord and don't understand where the sentiment is coming from. Also, some well-meaning folks don't really know how to season the truth with love, so they will say something like "I don't agree" but it will come out as "I hate gays" (like my bff... we were up until 3:00 a.m. Sunday morning talking about this... as much as she loves everybody and loves the Lord, her feelings toward gays is really borderline hatred - and I think if not for the God factor, it WOULD be hatred).
With those kinds of experiences, I think I can figure out how/why some radical gay people might misinterpret "I don't believe in...." as "I hate."
It's not very dissimilar to how some blacks would take it when white people say stuff like "I don't have anything against black people, I just don't believe the races should mix" or "I don't have a problem with the blacks, I just believe they should stay in their neighborhoods and go to separate schools and let us have ours." (Not comparing the issues, comparing how people receive such comments and why they take it differently from how the person may have meant it).
For the record, it's not all gays who think like that. My Twitter TL was full of gay church folks posting their Chick-fil-a pics in support of Dan Cathy.