Charles Thomson, secretary of the Continental Congress and a signer of the American Declaration of Independence coined the Latin phrase based on one of Virgil's poems, Virgin being an ancient Roman poet. Thomson believed it was a fitting poetic way of talking about the birth of a new nation founded upon the principles of individual liberty. A new era dawned upon the world with the founding of this new republic, and so a new world order, or a new order stepping upon the world stage of history--an era of human liberty build upon democratic and republican ideals. In conjuction with this Latin phrase, Thomson also came up with annuit coeptus "has favored", that Divine Providence has favored the undertaking of the creation of the fledgling new republic. Which is also why the Eye of Providence features on the Great Seal. The Eye of Providence is a common and recurring symbol that goes back centuries, originating as a symbol of God's omniscience.
Why the Eye of Providence? Historically Christianity has frowned upon depicting God the Father directly. It happens sometimes, but is technically not supposed to happen in Christian iconography. Going all the way back to the Second Council of Nicea in the 8th century which dealt with the Iconoclast Controversy, the Council affirmed that depicting saints and angels is not a violation of the commandment of making graven images; and that images of Christ, though He is God, is not a violation of the commandment because the Second Person of the Holy Trinity in becoming human has a physical image and likeness--He is human. The Council also affirmed that the Divine Essence cannot be depicted, God in His Essence is invisible and unknowable, and cannot be depicted. Therefore images of God the Father and God the Holy Spirit were technically not allowed, but images of Christ were allowed.
Christian artists, therefore, tended to find creative ways to symbolically or implicitly depict the Father and the Holy Spirit. In the case of the Holy Spirit Christian artists relied upon biblical imagery--such as the Holy Spirit described as a dove at Christ's baptism, or the fire of Pentecost. In the case of God the Father, an all-seeing eye was often used, to symbolically represent the Father's omnipresent and omniscient gaze upon all of history and the whole world. The eye often appears with a triangular halo, the three sides of a triangle being a direct reference to the Trinity (in the same way that Christ and symbols of Christ, such as the Agnus Dei, have a cross-halo pointing to His Cross).
The Eye of Providence, therefore, was a traditional way of representing Divine Providence in historic Western iconography and art, an easily recognizeable symbol. Even if many of those involved in the founding of America weren't Christians themselves (Many were Unitarians and Deists), and some like Thomas Paine were fairly antagonistic toward religion, the symbol of Divine Providence was still quite useful in saying that the creation of the new American republic was a monumentous event in the history of humankind.
So all of these feature on the Great Seal of the United States, the phrase Annuit Coeptis "Has Favored", Novus Ordo Seclorum "New Order of the World", and the Eye of Providence.
It is this Great Seal of the United States, official documents feature the Great Seal. And that includes bank notes produced by the Federal Reserve, aka "green backs", aka American paper money. Look on the back of a $1 bill, you'll see the reverse and obverse sides of the Great Seal, it even tells you what it is "The Great Seal..." "...Of the United States"
So, now you know why that Latin phrase is on the back of American currency. There's no conspiracy theory, the reason is open, obvious, there's no secret, no hidden agenda. It's just old fashioned patriotism.
-CryptoLutheran