George Takei does a masterful job of explaining why Kim Davis, the county clerk in Kentucky, is violating the First Amendment when she refuses to grant marriage licenses to same-sex couples.
The First Amendment has two clauses that are relevant here. One is the Establishment Clause, and the other is the Prohibition Clause. Congress may not prohibit free worship, and that is what so many claim, wrongly, is being violated. But it is also not empowered to establish any religion, nor to enact any laws favoring one religion over the other. Permitting a state employee to foist her religion upon others, denying them a fundamental right as articulated by the U.S. Supreme Court in Obergefell, would be to give government, through this agent, the power to impose religious doctrine and viewpoint. That it cannot do. Ms. Davis is in effect establishing religion by using her governmental powers to impose her religious views.