There are three connections between the Gettysburg Address and the Declaration of Independence.  All of these connections come from things that President Lincoln said in the address.  In each case, he alludes to the Declaration without actually mentioning it or quoting it.

First, Lincoln starts the speech by saying

Fourscore and seven years ago our fathers brought forth on this continent a new nation…

When he says this, he is referring to the signing of the Declaration of Independence.  That marked the founding of our country and it was 87 years before Lincoln spoke in 1863.

Second, Lincoln continues this first sentence of the speech by saying that the new nation was

conceived in liberty and dedicated to the proposition that all men are created equal.

When Lincoln says this, he is alluding to the Declaration because the Declaration says,

We hold these truths to be self-evident, that all men are created equal…

Clearly, when Lincoln says that the US was based on the idea that all men are created equal, he is referring to the words of the Declaration.

Finally, Lincoln ends by saying that the purpose of the Civil War is to make sure that “government of the people, by the people, for the people” continues to exist.  This is an allusion to the ideas of the Declaration.  The Declaration says that government only exists by the consent of the people.  This can be restated as saying that the government is “of the people” and “by the people.”  The Declaration also says that the point of government is to protect people’s rights.  This means that government is “for the people.”  In short, Lincoln is saying that the Civil War is being fought to uphold the democratic ideals of the Declaration.

In these three ways, the Gettysburg Address is connected to the Declaration of Independence even though Lincoln never specifically mentions that document.