In the name of the most holy and undivided Trinity. (Intro.1)

Politics was more of a religious matter back in the 18th century, and the idea of "separation of church and state" wasn't really a thing.

Also, Britain was technically a theocracy, with the head of state—the King—being also the head of the state church -- the Church of England. Fun fact: Jefferson was the one who came up with that idea, and he wasn't the most popular person in Britain at the time.

The next line is a lot more importance, as well as being considerably longer:

It having pleased the Divine Providence to dispose the hearts of the most serene and most potent Prince George the Third, by the grace of God, king of Great Britain, France, and Ireland, defender of the faith, duke of Brunswick and Lunebourg, arch-treasurer and prince elector of the Holy Roman Empire etc., and of the United States of America, to forget all past misunderstandings and differences that have unhappily interrupted the good correspondence and friendship which they mutually wish to restore, and to establish such a beneficial and satisfactory intercourse , between the two countries upon the ground of reciprocal advantages and mutual convenience as may promote and secure to both perpetual peace and harmony; and having for this desirable end already laid the foundation of peace and reconciliation by the Provisional Articles signed at Paris on the 30th of November 1782, by the commissioners empowered on each part, which articles were agreed to be inserted in and constitute the Treaty of Peace proposed to be concluded between the Crown of Great Britain and the said United States, but which treaty was not to be concluded until terms of peace should be agreed upon between Great Britain and France and his Britannic Majesty should be ready to conclude such treaty accordingly; and the treaty between Great Britain and France having since been concluded, his Britannic Majesty and the United States of America, in order to carry into full effect the Provisional Articles above mentioned, according to the tenor thereof, have constituted and appointed, that is to say his Britannic Majesty on his part, David Hartley, Esqr., member of the Parliament of Great Britain, and the said United States on their part, John Adams, Esqr., late a commissioner of the United States of America at the court of Versailles, late delegate in Congress from the state of Massachusetts, and chief justice of the said state, and minister plenipotentiary of the said United States to their high mightinesses the States General of the United Netherlands; Benjamin Franklin, Esqr., late delegate in Congress from the state of Pennsylvania, president of the convention of the said state, and minister plenipotentiary from the United States of America at the court of Versailles; John Jay, Esqr., late president of Congress and chief justice of the state of New York, and minister plenipotentiary from the said United States at the court of Madrid; to be plenipotentiaries for the concluding and signing the present definitive treaty; who after having reciprocally communicated their respective full powers have agreed upon and confirmed the following articles. (Intro.2)

Yeesh. That's one sentence.

And it basically just goes into who the two sides are. They list out all of King George's titles in a more that puts him head-to-head with Daenerys Targaryen in the "Royal With Longest Name List" contest

King George III is from the House of Hanover, which was a German noble house. That's right: a German was on the throne of England. Countries weren't thought of in the same way they are now, and, to the people at the time royal blood was way more important than nationality. (That explains why Hamilton briefly floated the idea of importing some monarchs from Europe to rule the United States.)

But yeah; those titles. There's a lot more than merely "King of England." Essentially, those boil down to the idea that for much of history, Britain thought it ruled, well, pretty much everything. This includes their ancestral enemy, France. Needless to say, France disagreed, and this claim ended up dropped when France became a republic in 1801.

So, to summarize, the opening lines basically mean the following. "Hey, we're all Christians here, and we're doing this with the blessing of our God. Over here, we have King George III and over here, we have representatives from the United States, which is a totally real country." It's important to know who a treaty is between before you hammer out specifics.