It just doesn't. The idea that those of us who consider same-sex activity to be immoral are somehow "anti-gay" bigots who diminish the value of their fellow human beings is, to put it bluntly, a load of bull.
Certainly, there are many people in this world who both consider same-sex activity to be sinful and also harbor distrust, animosity, and just plain hate for LGBT people. Prejudice is real and exists in many forms among many people today. This cannot be denied. But simple logic does not allow us to conclude that since some traditional morality advocates are bigots, all of them or most of them must be bigots. If one is to accuse these people of bigotry, it must be proven that there is something inherently bigoted in moral opposition to same-sex activity, and once the propaganda shoved in our face by the media is overthrown, it becomes clear that there is not.
Simply, believing another person's behavior to be immoral or wrong is not the same as harboring any hostility toward the person himself. Similarly, declaring that particular behavior to be wrong is not the same as verbally attacking a person who practices the behavior. This goes for any behavior, be it sexual, substance-related, or dealing with any other realm of ethics. Indeed, if it were necessary to be prejudiced against every person who practiced behaviors that, in your eyes, were immoral, you would have very few friends indeed!
Take smoking for an example. You or I might believe smoking to be wrong; "It is harmful to one's health," we might say, or, "It is bothersome to others." If we held this ethical position, we would most likely seek to abstain from smoking in our own lives. In conversations we might iterate our opinions on the matter; if we had close friends who were smokers, we might advise them to quit. However, we would not necessarily have any prejudice against people who smoked merely because of our opposition to smoking. We would probably still have friends who were smokers. We would not necessarily think less of them or harbor any resentment towards them because of their habit; we would simply realize that we could not affirm a certain behavior practiced by those friends, and that would be that. From this example, it is easy to see that one can criticize a particular behavior without attacking those who practice it.
However, this does not solve the whole problem, because it is not fair to LGBT people to compartmentalize their lives from the outside, identifying certain components of their lives as mere behaviors that do not influence our perception of the persons themselves. The problem is not merely one of behavior, but one of identity, and sexuality is a deeply ingrained component of human identity, tied to our deepest desires and our most ancient urges. Many heterosexuals do not often think about the place sexuality holds in their sense of self; it is easy to take for granted, just as it is easy to take other normal aspects of one's identity for granted. However, because those with homosexual orientations are in the minority in western civilization, and because these people have traditionally been marginalized and oppressed, and also because the issue of homosexuality is such a pressing one in today's culture, many LGBT people put a great focus on the sexual component of their identity, as has been the case with many other movements centered around minorities for centuries. From this perspective, it is easy to see why an argument against the morality of same-sex activity could appear to be an attack, not merely on a behavior, but on one's innermost identity.
Although these circumstances make the vehement reactions often seen towards moral conservatism more emotionally understandable, they still do not, from a rational perspective, delegitimize arguments against the morality of same-sex activity. For all behaviors, all beliefs, all urges, all desires are ultimately wrapped up in the identity of the individual who practices and holds them, and although some of these components of the self, depending on the disposition and situation of the person, are held and treasured more strongly than others, all play some part in the formation of the human identity; whether innate and inborn or nurtured or chosen, for a person is as much born as he is made and as much made as he is born. If every component of every person's identify is immune from criticism, then no moral statement can ever be made. Also, it is clear that a person's identity - or their perception of it - is not the same as another person's perception of him. Things about yourself that you hold very dear might be of no consequence to a friend; other things that you barely notice may bring you great endearment from others. This is not an evil thing but is merely a fact of life; no one can understand another perfectly and no one can fully embrace the whole of another's identify, for that would be to lose one's own individuality. Therefore, in order that humans might interact, we come to the understanding that we each may ignore, overlook, or critique particular components of the identity of another without, per se, doing the same to their person.
In order to illustrate this, another example will be helpful. Religious faith is a defining aspect of the identities of many people, just as much as sexuality defines the identities of others, and yet it would be ridiculous to suggest that thinking another person's religion to be false is somehow an attack on the person himself. A Christian and a Muslim, for example, might each consider another's faith to be founded upon lies but can still be friends with each other and engage each other in dialogue about religious and other issues. As a Christian, I believe Islam, Hinduism, Buddhism, Judaism, and all other faiths to be false religions, but that does not make me anti-Muslim, anti-Hindu, anti-Buddhist, or anti-Jew. Similarly, if someone were to tell me that they considered my religion to be false, I would not act as if they had insulted me but rather would attempt to engage their viewpoint. The same should be the case for discussing matters of sexuality. We should be able to differ on matters that involve our practices and behaviors, beliefs and opinions, and even the innermost cores of our identities, and yet treat each other with love, not defining others by their sexuality but by their status as fellow human beings (and, from a Christian perspective, as beings created in God's image) who deserve our respect.