Separation of church and state is a bedrock principle of our Constitution, much like the principles of separation of powers and checks and balances. In the first place, the Supreme Court has thoughtfully, authoritatively, and repeatedly decided as much; it is long since established law. In the second place, the Court is right. In the Constitution, the founders did not simply say in so many words that there should be separation of powers and checks and balances; rather, they actually separated the powers of government among three branches and established checks and balances. Similarly, they did not merely say there should be separation of church and state; rather, they actually separated them by (1) establishing a secular government on the power of "We the people" (not a deity), (2) according that government limited, enumerated powers, (3) saying nothing to connect that government to god(s) or religion, (4) saying nothing to give that government power over matters of god(s) or religion, and (5), indeed, saying nothing substantive about god(s) or religion at all except in a provision precluding any religious test for public office. Given the norms of the day (by which governments generally were grounded in some appeal to god(s)), the founders' avoidance of any expression in the Constitution suggesting that the government is somehow based on any religious belief was quite a remarkable and plainly intentional choice. They later buttressed this separation of government and religion with the First Amendment, which affirmatively constrains the government from undertaking to establish religion or prohibit individuals from freely exercising their religions. The basic principle, thus, rests on much more than just the First Amendment.
The original pilgrims who came to North America from England sought freedom, including religious freedom. They didn't like the idea of having to subscribe to government sponsored religion. Some of the Founding Fathers were deists instead of strictly Christians and did not like the idea of the United States establishing a certain religious interpretation over others.