The iconic status of the Declaration is due to several factors involving the inherent qualities of the document, the historical context in which it appeared, and the subsequent history of the world.
It's ironic that most of the document deals with specific charges made against the British king rather than the universal precepts it enunciates. In my view the essential passages are the first two paragraphs, and the last paragraph. The Declaration states simply and directly the liberal worldview that people are equal and free, and that no government has the right to control others by interfering with their freedom or with their attempts to act as they choose and to flourish. This brief paraphrase lacks the eloquence of Jefferson's prose, and this leads us to the second factor about the document's greatness: it's well written, in a supercharged, poetic style.
Just as with Shakespeare, the way ideas are expressed, the words chosen, are as important as the ideas themselves. The principles espoused by Jefferson and Congress were not new. The liberal ideas of government were expressed by the Enlightenment philosophers from John Locke forward. But the Declaration is an encapsulation of Enlightenment thinking, a microcosm of the new way in which mankind had begun to view itself. The crucial juncture of that moment in time, in which progressivism began to be put into practice, endowed the Declaration with the semi-mythic status it still holds today.
Had the history of the world gone differently at this point, posterity's way of viewing these words could have been quite different. Thirteen years to the month after the document was completed and approved, the French Revolution began. The fact that France had facilitated American independence, the fact that Jefferson was present in Paris in 1789 and had a close friendship with French revolutionary Lafayette, and the fact of the Declaration as the model of a written statement that would serve as a democracy's founding document, all worked together to transform it from a merely local and specific list of grievances (overlaid with a philosophical message) into a universal creed for humanity. Most people would agree mankind is in a better place collectively now, despite its still massive flaws, than it was 230 years ago. Eventually slavery was eliminated in America, serfdom eliminated in Europe, and the old ancien régime of the European political system would be superseded by democracy or quasi-democracy, despite the periodic setbacks that occurred once and again, like dips in an overall ascending line on a graph.
Admittedly this is a narrow way of conceptualizing history, but it's at least partly true that the improvement that has occurred in the world, despite the enormous flaws and the catastrophes that have continued into our own time, is a real-life projection of the ideals stated in the Declaration of Independence. The Declaration provided the new country "brought forth on this continent" with a kind of thesis regarding what America, and by extension the world, would base itself on, regardless of how imperfectly its ideals have been achieved in practice.