If you're asking "how" in the practical sense, as in, through what means did they frame their side's struggle as analogous to the Founding Fathers, it was a slum dunk for both sides, really.
In the north, Lincoln and the Union could (and did) claim that they were defending the nation the Founding Fathers had forged through blood, sweat, and sacrifice. Further, they could (and again, quite often did) claim that they were fighting to defend the idea of a free republic, that they were working to see the "great experiment" endure. In this way, the Union could argue that they were taking up the work of Washington, Jefferson, and Adams (etc) by "minding the store" if you will. Soldiers and supporters of the Union were fighting to preserve The United States of America, after all, which is the country the Founding Fathers founded!
Those Founding Fathers didn't create the Confederate States of America, so the rebels had to frame their claim of the Founding Father imagery via a narrative hijacking. The Confederates played up the right to self-governance and personal liberty angle, and put the Union in the role of Great Britain. Like Washington et al, the south claimed that they were and ought to be seen as an independent nation capable of self-governance. Jefferson Davis in particular liked to give speeches and make proclamations that framed the Civil War as a second war of independence for the south, and that he, like George Washington, was leading a valiant struggle against a former host-nation that no longer represented their interests.
At its most basic level, this analogy kinda holds, as the Confederacy was indeed a part of the U.S., just like the colonies had been a part of G.B. To further play into this dramatic narrative, Davis and the Confederacy wrote up a Constitution, formed a government, and generally tried to go through similar motions that had worked for the Founding Fathers.
So there you have it. It's hard to cite anything in particular for this response, I might just start with some good, well-regarded writing on the subject such as James McPherson's Battle Cry of Freedom, or my personal favorite, Bruce Catton (any of his Army of the Potomac books are great reads).
The southern states used the writings of one of the founding fathers as part of their justification for secession. The Kentucky Resolutions of 1798, written by Thomas Jefferson, starts with "That the several States composing, the United States of America, are not united on the principle of unlimited submission to their general government; but that, by a compact under the style and title of a Constitution for the United States . . " and they setup justification for states to leave the Union under certain circumstances.
South Carolina and some of the other states quoted from the Declaration of Independence to also justify secession "that they are, and of right ought to be, FREE AND INDEPENDENT STATES; and that, as free and independent States, they have full power to levy war, conclude peace, contract alliances, establish commerce, and to do all other acts and things which independent States may of right do."