The citizens of the thirteen colonies were British citizens, but America was a very different world than Great Britain—and we're not just talking about the fact that people in the British Isles think "hot summer day" means "66 degrees and only partially cloudy."
Britain in the 18th century had a few very, very rich people (think Warren Buffet, but richer), and lots and lots of very poor people. The colonies, in contrast, were much more egalitarian, with many more landowners and far less extreme poverty. Basically, if you made the journey and left the mother country, you could probably own your own land and save money much more easily in the colonies.
No wonder the colonists were more inclined to think that "all men are created equal"—in their world, that was much closer to reality. (Let's hear it for the colonies, y'all.)
Not that it was all utopian sunshine and roses (after all, slavery was already established), but overall the social structure of the colonies heavily impacted how the colonists viewed themselves in relation to England. They saw their society and their governmental system as what England's should be.
Colonists were still fascinated with the mother country—who isn't into cute English accents, scones, and badgers?—but the people in England generally thought of the colonies as an uncivilized backwater (if they thought about them at all, that is).
It was as if Britain was Regina George and the Plastics—that group of popular kids that everyone sort of hates but also wants to be. Parliament made little effort to control the colonies, a system known as "benign neglect," leaving the American colonists to basically control themselves.
The Declaration of Independence was the culmination of years of dissatisfaction in the American colonies. People had been grumbling for a while, but the real anger started most clearly around 1763, at the end of the Seven Years' War (which is often referred to as the French and Indian War in the American context).
Colonists after the war believed all that land won in the war would be available for them to settle, and their contribution would be recognized and result in lower taxes. However, with the debt of the war, the government needed to make the colonies as profitable as possible, so literally the exact opposite happened. And you can imagine how happy the people of the colonies were about that.
The British government started doing a bunch of things that felt pretty unfair to the colonists, who were, you'll remember, 100% British citizens. For example, after the war, the British government left thousands of soldiers in the colonies, in theory to protect against rebellion by Native Americans, but also to stop the colonists from settling further west. (Source)
But the biggest issue for a while was taxes. American colonists weren't allowed any representatives in British Parliament, yet they were paying taxes to the British government. That's just unfair. Here's a lesson for the ages: when you have a constitution, taxation without representation does not go over well.
Parliament then started adding on taxes to the colonies, to tighten control and make the region more profitable, since the English national debt had skyrocketed.
The most famous is the Stamp Act, which required colonists to use special stamped paper for nearly everything—oh, and they had to pay extra for that paper. That's right, they were legally required to buy a specially marked paper (and dice, randomly) to use for a whole mess of things.
No stamp? Off to military court with you! Hope you don't want a jury…
In reality, the tax wasn't a huge financial burden on the colonists, but it was the principle of the thing (*thumps fist on table). Parliament was getting all up in the colonies' business, and the colonists worried about where that newfound nosiness could lead. (Source)
American colonists began to resist. Beginning with Virginia, colonial assemblies claimed the that they alone could tax their citizens, and groups known as the "Sons of Liberty" sometimes took, ahem, aggressive action to show their disapproval. (Sometimes fire was involved.)
By 1769, most of the colonies had agreed to an embargo on British goods, an idea thought up by Samuel Adams. (Who was, despite what the first three results on a Google search of "Sam Adams" might lead you to believe, a man and not just a brewery.)
Many of the most oppressive taxes were repealed by 1770, but in 1773 Parliament passed the Tea Act, leading to the famous Boston Tea Party—led again by Samuel Adams, who was a very busy man.
Generally people date the beginning of the American Revolution, or the Revolutionary War if that suits your fancy, to the Battle of Lexington and Concord on April 19, 1775. By August of 1775, King George III declared the colonies in open rebellion against England. From that point on, the colonies really started pulling away from Britain…which is maybe not surprising given the king's declaration. He didn't exactly roll out the welcome mat and tell them to pull up a chair.
Finally, on June 7, 1776, the Second Continental Congress passed the Lee Resolution, named for Richard Henry Lee, which said that the colonies should be independent. After that, Jefferson and the Committee of Five were appointed to draft the Declaration of Independence. The document was presented on July 2, 1776, edits were made, and the final version approved on July 4th.
Those lofty philosophical ideas that Jefferson includes in the Declaration didn't just come from a super-trippy-yet-inspirational dream he had one time. The idea of government being responsible to its people and for protecting their rights as humans came out of the Enlightenment and specifically John Locke's treatise on government.
You can also see the idea reflected in the U.S. Constitution and the Bill of Rights, which both sought to rectify what the colonists saw as the tyranny of Great Britain by creating a government formed, as Abraham Lincoln said in the Gettysburg Address, "by the people, for the people."
And, bonus: Jefferson wasn't the only one reading Enlightenment philosophy and incorporating it into his work at this time. Even if philosophy and pondering the meaning of our existence isn't your thing, some very important people in history were really into it, and it has probably had a tangible impact on your life.