To the founding fathers, the separation of church and state meant freedom of conscience. You can get more of an idea of what Thomas Jefferson meant by it in looking at his fight to disestablish religion in Virginia:
Well aware that the opinions and belief of men depend not on their own will, but follow involuntarily the evidence proposed to their minds;
That Almighty God hath created the mind free, and manifested his supreme will that free it shall remain by making it altogether insusceptible of restraint;
That all attempts to influence it by temporal punishments, or burthens, or by civil incapacitations, tend only to beget habits of hypocrisy and meanness, and are a departure from the plan of the holy author of our religion, who being lord both of body and mind, yet chose not to propagate it by coercions on either, as was in his Almighty power to do, but to extend it by its influence on reason alone; [...]
We the General Assembly of Virginia do enact that no man shall be compelled to frequent or support any religious worship, place, or ministry whatsoever, nor shall be enforced, restrained, molested, or burthened in his body or goods, nor shall otherwise suffer, on account of his religious opinions or belief; but that all men shall be free to profess, and by argument to maintain, their opinions in matters of religion, and that the same shall in no wise diminish, enlarge, or affect their civil capacities.
Here you have an illustration of true separation of church and state. Note that it is not a separation of God from government, for it’s logic support beam is that “Almighty God hath created the mind free” and therefore it should be free. No one shall be forced to support a religion they do not believe in, etc…
Basically they wanted to get away from the European model where being Anglican or Catholic or whatever was so wrapped up into politics. They wanted you to be free to worship according to the dictates of your own conscience, or to not worship at all. So there is no religious test to run for office, etc…
To that end they established no religion. However, this did not mean that you could never mention God in the public square. To say that it did is to take away freedom of speech and religion. Extremists have redefined the separation of church and state to mean things like you can’t bring a Bible into school.
Christians sometimes go overboard with the whole “there is no separation of church and state!” thing. Of course there is. The church shouldn’t run the state and the state shouldn’t run the church. They have different functions. Personally, I’m against government money going to faith-based institutions. I was always taught that government money = government control, and the church should stay far away from that. I also believe that the state is a corrupting influence and the more married the church is to it, the more corrupt the church becomes. At some point it ceases to be about carrying out the commands of Christ and becomes about politics and power. It leads to horrors being done in His name.
Alexis de Tocqueville, Democracy in America, Part I:
The philosophers of the eighteenth century explained the gradual decay of religious faith in a very simple manner. Religious zeal, said they, must necessarily fail, the more generally liberty is established and knowledge diffused. Unfortunately, facts are by no means in accordance with their theory. There are certain populations in Europe whose unbelief is only equalled by their ignorance and their debasement, whilst in America one of the freest and most enlightened nations in the world fulfils all the outward duties of religious fervor.
Upon my arrival in the United States, the religious aspect of the country was the first thing that struck my attention; and the longer I stayed there the more did I perceive the great political consequences resulting from this state of things, to which I was unaccustomed. In France I had almost always seen the spirit of religion and the spirit of freedom pursuing courses diametrically opposed to each other; but in America I found that they were intimately united, and that they reigned in common over the same country.
As a member of the Roman Catholic Church I was more particularly brought into contact with several of its priests, with whom I became intimately acquainted. To each of these men I expressed my astonishment and I explained my doubts; I found that they differed upon matters of detail alone; and that they mainly attributed the peaceful dominion of religion in their country to the separation of Church and State. I do not hesitate to affirm that during my stay in America I did not meet with a single individual, of the clergy or of the laity, who was not of the same opinion upon this point.
As long as a religion rests upon those sentiments which are the consolation of all affliction, it may attract the affections of mankind. But if it be mixed up with the bitter passions of the world, it may be constrained to defend allies whom its interests, and not the principle of love, have given to it; or to repel as antagonists men who are still attached to its own spirit, however opposed they may be to the powers to which it is allied. The Church cannot share the temporal power of the State without being the object of a portion of that animosity which the latter excites.
The unbelievers of Europe attack the Christians as their political opponents, rather than as their religious adversaries; they hate the Christian religion as the opinion of a party, much more than as an error of belief; and they reject the clergy less because they are the representatives of the Divinity than because they are the allies of authority.
In Europe, Christianity has been intimately united to the powers of the earth. Those powers are now in decay, and it is, as it were, buried under their ruins. The living body of religion has been bound down to the dead corpse of superannuated polity: cut but the bonds which restrain it, and that which is alive will rise once more. I know not what could restore the Christian Church of Europe to the energy of its earlier days; that power belongs to God alone; but it may be the effect of human policy to leave the faith in the full exercise of the strength which it still retains.
In the United States the sovereign authority is religious, and consequently hypocrisy must be common; but there is no country in the whole world in which the Christian religion retains a greater influence over the souls of men than in America; and there can be no greater proof of its utility, and of its conformity to human nature, than that its influence is most powerfully felt over the most enlightened and free nation of the earth.
I have remarked that the members of the American clergy in general, without even excepting those who do not admit religious liberty, are all in favor of civil freedom; but they do not support any particular political system. They keep aloof from parties and from public affairs. In the United States religion exercises but little influence upon the laws and upon the details of public opinion, but it directs the manners of the community, and by regulating domestic life it regulates the State.