They were fighting for self government...for the principle that government derives its legitimacy from the consent of the governed.
The states never agreed to surrender their sovereignty and bind themselves forever when they ratified the constitution. Indeed 3 states including Virginia expressly reserved the right to unilateral secession at the time of ratification.
They were fighting to rip the United States apart, plain and simple. Many Southeners would not go along with that and like Trump’s hero Amdrew Jackson would have, they supported the Union. The South was not all Confedrates during the War.
Does "the governed" also refer to the enslaved?