I fear, however, that my eloquence fails me.
Majority rule; minority rights.
Nothing could really sum of the principles of American Democracy better, nor in particular, the separation of church and state.
The American democratic experiment is not one in pure democracy. Why not? Because when the Founders looked at democratic governments of the past, they saw that democracies were fragile forms of government that were subject to the whims of the majority, or, put more bluntly, the mob. Both Greek and Roman popular governments descended into tyranny of one sort or another on the support of a majority of the people. Whether an injustice is committed by one despotic ruler or by a government elected by an unchecked majority, it matters not all in determining whether the underlying act is unjust or not. When a person's rights are denied to them, rights that are granted to others, the source of the evil is irrelevant. Thomas Jefferson argued vociferously against the dangers of mob rule, and whether or not we agree with his particulars or not, the principle remains a sound one.
If America were simply a democracy, there would be very little need for a Constitution. There would be even less need for a Bill of Rights. After all, if the majority could do no wrong, what would be the point? Instead, our Constitution lays out not only the form of government (a democratic repulic), but also the restrictions that government must operate under. It lays out both principles of American democracy: majority rule, but with specific protections for the rights of the minority of citizens and how those rights are secured through the Judiciary. If the majority wishes to stake a claim to a right for itself, it must grant that right to everyone. The question of religion follows this general pattern.
The majority Christians wish to claim the right to practice their beliefs as they see fit. If they are to practice that religion, then they must accept that others will practice (or not practice) religion as they see fit. It is no right at all to say that one has a right to be Christian. Each of us may follow the dictates of our own conscience, restrained only by reasonable restrictions from the state to which everyone is subject, and which can be justified in the general public interest.
There is a limit to one's ability to practice one's religion, however. In one's private life, if a religious person chooses not to be friends with an atheist, that is their right, and mine. But, they do not have the same right to disassociate from those who disagree with them in the public sphere. Even Jesus, ironically, teaches that religion should be the province of one's own conscience and not worn on one's sleeve to impress others or win their favour. In the public sphere, the Constitution is also very clear. It states explicitly that there will be no religious test for office (and since god belief is a religious one, even atheists should be able to get elected (in a perfect world)). The First Amendment likewise specifies that no religion will be favoured. This means not only that Baptists can't be favoured over Catholics, it means that no religion, or even religion itself, can be favoured over any other, or even no religion at all.
To believe that it means otherwise is to violate the very principle of majority rule, minority rights. For if the minority of the citizens disbelieve in god, how does even favouring mere religion over irreligion protect their right to free exercise of their conscience? Just believe in something? Who gets to determine what is and is not an acceptable "something"? Nothing in the Constitution suggests that thought crimes are within the province of just government.
Governments rarely observe the principles they espouse perfectly. It is for this reason that we must also seek to embody those ideals better than we do today. It should always be a striving forward to do better than before, never to justify our current practices by "tradition" or because that is the best that previous generations could do. But these are exactly the arguments made by those who would maintain or increase the role of religion in the public sphere. They look to the past, rather than to the principles of our democracy.
As an atheist, of course, I have a vested interest in freeing my government from the shackles of every last trace of religion. The principle, of course, matters, but when I have to wonder how my government might use my hard-won atheism against me--when some judge may decide I may actually be discriminated against in employment, or that I may have to send my future children to a public school that seeks to indoctrinate them in a/n un/conscious belief that their mother is unpatriotic at best, and a monster at worst, or uses my disbelief against me in court, or a host of other "inconveniences"--then my government has lost sight of the principles it was founded on, and sooner rather than later, it will not just be I who am wondering these things; it will be those of every minority belief as well. When conformity becomes a matter of public policy, no one is free.
However, seeking to remove religion from government, or even the public sphere, does not equate to promoting atheism, or even official agnosticism. Rather, separation of church and state (representing the public sphere) seeks to make the state officially silent on matters of religion. The state should neither hinder nor promote religion of any kind... not even "ceremonial deism". When government remains silent on such questions, it can no more promote religion than it can promote irreligion.
Of course, this is all a matter of government. In the private sphere (and by that I mean non-governmental sphere here), the "respect" for religion and irreligion promoted by separation of church and state does not mean that I cannot argue with those outside the confines of government coersion. Individual knowledge does not increase without vigorous debate. As long as I am not using my position in government or public accomodation to oblige others to conform to my viewpoint, then discussion is encouraged and truly essential, particularly when it can be free of the arrogant presumption that one side is correct regardless of the facts that might be brought to bear on the discussion. Some arguments will never be resolved satisfactorially, but then, that is life. Deal with it. :)