I had a discussion with one of my conspecifics some time ago about the "separation of church and state". He insisted that these words are nowhere to be found in the Constitution. True enough. The Establishment Clause and the Free Exercise Clause are in the Bill of Rights, and these are the two provisions of the Constitution which give rise to the doctrine of Separation of Church and State.
How else can the government avoid running afoul of the two clauses about religion? It can't establish an official religion, and it can't regulate your irrational beliefs about the supernatural. Anything the government does to entangle itself with churches or religious ideas or movements is bound to have the effect of privileging or prejudicing, and that amounts to establishment or interference with free exercise. It's best to keep the government as far from religion as is practicable.
Of course, government is run by folks with religious beliefs, and they will likely govern their own behavior in accordance with their consciences, so religion will necessarily play a role, albeit indirect, in the functioning of the state. It's best not to make a big show of it, though, because carved in stone religious pronouncements are inconsistent with a democratic polity. At some point, one set of normative propositions predicated on a supernatural premise will lose out to a competing set of propositions, and the adherents of the former set will have to be prepared to live with it. Otherwise, sectarian strife ensues. The differences among Protestant denominations are quite sufficient to set Christians to murdering one another if you put religion and politics together too much. God forbid that the religious right should ever get its wish to establish a theocracy, because there would be a bloodbath over the precise nature of theo before too long.