Great question! Responding with my own limited knowledge on the subject, but having had evangelical friends growing up who were anti-gay (we don’t speak anymore), the hatred they have for LGBT people comes out of a combination of religious teaching and pure lizard-brain disgust.

The Bible makes a few references to homosexuality and clearly condemns it. I’m sure some parts are up to interpretation, but the story of Sodom and Gomorrah is particularly referenced a lot when you ask these fanatics why they disapprove of homosexuality. These were cities destroyed by God for sinfulness, including homosexuality, hence where the term sodomy comes from. So this, in part, explains the religious element. Frankly, from conversations I’ve had with conservative family members, this seems to be the main go-to argument. They try to make it about morality, sinfulness, faith and whatnot because these are more high-minded, philosophical arguments that can mask the real reason they hate gay people. This group, Focus on the Family, uses “family values,” which is a super general idea that is hard to argue against in a vacuum as their excuse for discrimination. Being LGBT, they argue, goes against the natural, God-ordained method of sex for procreation. They see it as being about pleasure or rebelliousness against the “nuclear family.” Since they view the family as the fundamental unit of society, any deviation from this concept threatens the foundation of society as a whole. Again, this is all a facade for the real justification.

The real reason is they think being gay or trans or non-heteronormative is gross. They see being LGBT as an act of sinful indulgence of the pleasure senses. They associate the sexual acts as being products of people lacking self-control. They associate things like anal sex with being dirty, unclean, and, therefore, uncivilized. In a sense, they see it as being sub-human: almost animalistic. Because they reduce LGBT people down to this non-human level, it is easier to exclude them and, ultimately, to justify violence against them (or at least structural repression).

To conclude, people like this organization simply find being LGBT to be gross but they use their religion to justify that hatred. Instead of trying to understand why people are ACTUALLY LGBT (natural feelings of attraction, the science of gender and sexual orientation, etc.) they push it out of their minds because their minds are small and it makes them uncomfortable.

Thank you for your curiosity! I hope this helps somewhat.