Most specifically, it was intended to keep the state from mandating it's citizens to practice a particular religion, or to outlaw another. Many of the original colonists were victims of religious persecution who came to the colonies in order to have freedom to worship as they chose.
Well said, it was never the intention of the founding fathers to make government secular. Divine providence, and the moral obligation found therein, was meant to be a part of our society and culture.
As for George Washington's quote, here is what he really said (I like it even better) ...
"Of all the dispositions and habits which lead to political prosperity, religion and morality are indispensable supports. In vain would that man claim the tribute of patriotism who should labor to subvert these great pillars of human happiness, these firmest prop of the duties of men and citizens. The mere politician, equally with the pious man, ought to respect and to cherish them. A volume could not trace all their connections with private and public felicity. Let it simply be asked: Where is the security for property, for reputation, for life, if the sense of religious obligation desert the oaths which are the instruments of investigation in courts of justice? And let us with caution indulge in the supposition that morality can be maintained without religion. Whatever may be conceded to the influence of refined education on minds of peculiar structure, reason and experience both forbid us to expect that national morality can prevail in exclusion of religious principle... Observe good faith and justice toward all nations. Cultivate peace and harmony with all. Religion and morality enjoin this conduct; and can it be that good policy does not equally enjoin it?" George Washington, Farewell Address, To his cabinet, Philadelphia, Pennsylvania; September 17, 1796.
I believe Washington made it clear that "religion and morality" are "indispensable supports" of government ... and these words were said to the "governing" and not to the "governed".
