Following the announcement of the Declaration on July 4, 1776 and the eventual parchment version being signed on August 2, 1776, the document itself was neglected after the revolution. Even early celebrations of Independence Day ignored the original statement of that independence. It was the act that was thought important, not the text.
The Declaration became important again after the founding of political parties. Once that happened, Jefferson's supporters used the fact that he wrote it to their political advantage. This created heated back and forth over the document's authorship itself and eventually resulted in it being more prominently thought of in terms of importance of the text. Even then, however, it wasn't until the 1850s that the document itself became important for more than historical reasons.
Once the abolitionist movement gained prominence in the early 1850s the "all men are created equal…" concept embedded in the Declaration became useful in the struggle to end slavery in the United States.
This usage of the text was taken up by Abraham Lincoln in 1854. He felt that the Founding Fathers expected that slavery would be a dying institution in the new United States. Lincoln also felt that the Declaration of Independence was one of the founding documents of the nation and not just a simple statement declaring secession from Britain. He used this view frequently in his arguments against slavery:
"Nearly eighty years ago we began by declaring that all men are created equal; but now from that beginning we have run down to the other declaration, that for some men to enslave others is a 'sacred right of self-government.' … Our republican robe is soiled and trailed in the dust. Let us repurify it. … Let us re-adopt the Declaration of Independence, and with it, the practices, and policy, which harmonize with it. … If we do this, we shall not only have saved the Union: but we shall have saved it, as to make, and keep it, forever worthy of the saving."Lincoln's view that the Declaration was one of the founding documents in terms of defining the nation eventually became the nation's view, even though it was not predominately so before him. This was an extremely important development in America's history in terms of interpreting the Constitution.
You can read more about the history of the document itself here.