The First Amendment says that "Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof...". There is an obvious problem here: are laws against murder "prohibiting the free exercise" of the Religion of Murder? In 1878, the Supreme Court held that laws against polygamy did not violate the First Amendment rights of Mormons, under the reasoning that the First Amendment protected religious beliefs but not practices.
In the 1960s, the Warren Court did a lot of stuff, and in particular it took another look at the Free Exercise clause of the First Amendment. In Sherbert v. Verner, the plaintiff, Sherbert, was a Seventh-day Adventist who was fired after refusing to work on the Sabbath. She was denied unemployment and sued. The Court ruled in her favor, setting out a three-prong test to determine whether the government violated an individual's Free Exercise rights:
a "compelling state interest" justifying the infringement (the "compelling state interest" prong), AND
no alternative form of regulation that can avoid infringement and achieving the state interest? (the "less restrictive means" prong)
This test (commonly seen in civil rights cases) is called "strict scrutiny". It's not easy for government laws to pass this test. This made religious people very happy.
In 1990, this changed dramatically with Employment Division v. Smith. (I'm simplifying the facts of the case here; you can Wikipedia it to see its unusual procedural history.) The question for the Court was whether an Oregon law against peyote violated the Free Exercise clause when the Native American Church commonly ingested peyote as part of a religious ceremony.
The Court found that the law was OK, because it was not specifically aimed at a physical act engaged in for a religious reason. It was a neutral law of general applicability.
Smith argued that wait, doesn't Oregon have to show a "compelling state interest"? The Court disagreed. It said that in those previous unemployment cases, the government had to show a "compelling state interest" in denying those people in particular their unemployment benefits. Here, on the other hand, the law applies to everyone equally without room for individualized consideration.
The "compelling government interest" requirement seems benign, because it is familiar from other fields. But using it as the standard that must be met before the government may accord different treatment on the basis of race, see, e.g., Palmore v. Sidoti, or before the government may regulate the content of speech, see, e.g., Sable Communications of California v. FCC, is not remotely comparable to using it for the purpose asserted here. What it produces in those other fields -- equality of treatment, and an unrestricted flow of contending speech -- are constitutional norms; what it would produce here -- a private right to ignore generally applicable laws -- is a constitutional anomaly.
… The rule respondents favor would open the prospect of constitutionally required religious exemptions from civic obligations of almost every conceivable kind -- ranging from compulsory military service to the payment of taxes to health and safety regulation such as manslaughter and child neglect laws, compulsory vaccination laws, drug laws, and traffic laws; to social welfare legislation such as minimum wage laws, child labor laws, animal cruelty laws, environmental protection laws, and laws providing for equality of opportunity for the races.
The Court said that if you wanted relief from these laws, you should be seeking redress in the legislature. Have them carve you out an exemption.
Congress immediately passed the Religious Freedom Restoration Act of 1993. It basically said, "Screw you Employment Division v. Smith, go back to Sherbert v. Verner and strict scrutiny." In particular, the law says "Government shall not substantially burden a person’s exercise of religion even if the burden results from a rule of general applicability."
The Supreme Court did not strike this down. However, it said that Congress couldn't go so far and pass this law on behalf of every state government. It could only do this with respect to federal laws.
So now a bunch of states passed RFRA's.
In 2012, Burwell v. Hobby Lobby Stores, Inc. threw another wrench into the whole system. To vastly oversimplify again, "ObamaCare" requires employers to provide contraceptive care, but has some religious exemptions for churches, nonprofit corporations, and very small employers. A closely-held corporation asked: wait, why don't I qualify for these religious exemptions? I believe life begins at fertilization, and don't want to have to pay for Plan B. Following the federal RFRA framework, the Supreme Court found that a closely-held corporation counts as a "person" entitled to religious beliefs, and that there was a "less restrictive means" of accomplishing whatever it was that the government wanted to accomplish.
Now we come to Indiana. Prior to Hobby Lobby no state RFRA really included corporations in their list of persons with First Amendment religious rights. Indiana has explicitly expanded their RFRA to include basically any corporation, even though Hobby Lobby was limited to closely-held corporations. And the other huge twist to this is that the Indiana RFRA allows a person/corporation etc. to assert this as a defense against private lawsuits, even when the government isn't involved. This is dramatically different from every other state's RFRA.
So there you have it. RFRA is not explicitly an anti-gay bill. On the other hand, Indiana's RFRA is ridiculously broad and (theoretically) allows a company to get away with a lot if they have a religious motivation -- which, in today's society, basically means anti-gay stuff.
In many respects it involves the same old uncomfortable question: what's the philosophical justification for civil rights bills that ban restaurants from anti-black/gay/female/etc. policies? We all agree that banning discrimination is a good thing, but why doesn't the business have the freedom to decide what it wants to be doing? Except that this debate is even more volatile because if you're anti-black people, you don't really have any leg to stand on, but if you're anti-gay, you can hide behind the curtain of your religious beliefs.
Finally, the Toxoplasma of Rage correctly predicts that the only reason this particular statute is getting so much attention is that it's neither unequivocally good or bad. There are a lot of complicated background moving parts and anyone who tries to give you a short sound bite as to why this is destroying this or that is either deliberately misleading you or just misinformed and closed-minded.