One piece of relevant history is a case known as Employment Division v Smith; sometimes called the "peyote case". Here SCOTUS ruled - in an opinion written by Justice Scalia - that the government was allowed to make laws barring Native American religious practices that involved ingesting peyote. The conservative justices joined Scalia in the majority and were opposed by liberal justices, who believed that the Constitution protected minority religious practices despite government interests in things such as drug regulation.
So why have things (apparently) flipped now? My opinion is that a great deal revolves around that notion of "minority". Liberals (very broadly speaking) differ from conservatives in their view of how Christian religious practices are handled in America. The (in my view fictitious) "war on Christmas" is emblematic of this difference. Conservatives see their religion under attack; liberals see an attempt to stop the majority from forcing its practices on everyone else. In Hobby Lobby, the owners of the company wish to require their employees to follow their own personal religious beliefs, and not access contraception. In Employment Division, the government wished to trump a minority religious practice with a majority view that peyote should be illegal.
There's a parallel here, I think, with responses to gay marriage. Conservatives cast efforts to promote same-sex marriage as an attack on themselves; liberals cast it as an attempt to get equal treatment for a minority. This case also implicates abortion, because at least Hobby Lobby has admitted that it is not opposed to all contraceptive coverage, only to coverage that its owners see as providing abortion-like services. Abortion cases often push people toward traditional positions.
My personal opinion is that if Hobby Lobby wins, even narrowly, it will be a spectacularly bad thing. It will be the first time that the Court has been willing to equate a corporation with its owners in matters of religion, and it would open up pretty much every piece of legislation and regulation to religious examination. And it would (further) damage important freedoms the First Amendment ought to be giving us.