If memory of past years and happenstance reading occasions serve, the more foresighted Christian philosophers, LGBTQ (or the like) activists, and Constitution-watchers have long predicted a win-or-lose clash in the US between right of conscience on the one hand--typically religiously motivated, usually of the Christian religion in the US--and the rights of, shall we say, sexual expressions outside the traditional on the other (on what basis do we now exclude inappropriate behavior with animals and incest?).
Partly the prediction merely leans on events in Europe which appear to be further along than on this side of the pond, but there are other contributing reasons from within US context itself over the decades and generations arguably such as with regard to the progress of divorce, abortion, femininism and egalitarianism (that is varied expressions thereof) and what may broadly be termed the sexual revolution and moral relativism and post-modernism and secularization (never mind overlaps). The recent Indiana frenzy has old and varied roots.
The right of one pizzeria owner in Indiana not to serve a potential self-identifying gay customer is, I think, more of a fringe distraction feeding the frenzy rather than constituting a good example of the hard legal and social decisions affecting large swaths of the citizenry.
And in my understanding at least, the hard decisions are often subtle from a Christian perspective. What is intended and being communicated (and not communicated) by baking or refusing to bake a wedding cake for a gay couple? Is the position of a baker employee told to bake the same as that of the employer? For example. In some cases it may come down to individual conscience and ill-understood context even if broad principles are present.
And as a Christian, my theologically-driven response (and to a certain extent experientially driven) includes a suspicion that many are not in a good position to listen to some of the subtle points--including ones central in the opinion of the Christian under fire in such cases.
How does one argue a case in which the arguments have been "ruled out of court"--the court, that is especially, of public opinion? Or where there is a listening ear, when what is fundamental and weighty in the eyes of the Christian--something to stake one's job, reputation, or civil freedom on--is seen as mere flighty opinion of reprehensible sort? Or when the mere claim that homosexuality is a sin in itself indicts the Christian as a homophobe or anti-homosexual bully or narrow-minded bigot? Either homosexuality is a moral evil or the judgment that it is a moral evil is evil. Winner takes all (even if that is somewhat of an exaggeration across the population).
In my opinion, what is particularly at stake in the long haul for the Christian is the right to say what is right--what is demanded by God to say. That this in particular is what is vilified or ignored or marginalized (despite the biological demands of reproduction and the testimony of chromosomes?) does seem to suggest anger at, and I think as Paulos23 rightly puts it, fear of Christians--perhaps fear lest the Christian (or perhaps substantially yet with differences, US) traditional perspective be taken at all seriously on the matter.