During Reconstruction, the Southern white population waged a violent campaign against franchising the freedmen. Sometimes it was organised, like the KKK. Sometimes, it was more incidental, as when just the sight of a unit of black troops in North Carolina caused a murderous riot. The Grant administration initially supported the Freedmen's Bureau and would respond to such attacks by dispatching troops. But the Panic of 1873 ( an economic depression) focused the rest of the US on the cost of military operations in the South, and Grant's efforts became much more sporadic. He himself became more cynical, disillusioned by the corruption he discovered in the friends he appointed to his cabinet, and began to talk hopefully of sending all the freedmen to what's now the Dominican Republic instead of preserving their rights in the US, and allowed the Freedman's Bureau to be obliterated. After they were able to regain their own rights to vote, the white Southerners also were able to create a strong Democratic political faction in Congress . After the Hayes/Tilden election dispute in 1877, they were able to exact large concessions from the Republicans in exchange for overlooking the Republicans' theft of the election. This effectively ended Reconstruction, and the Southern Democratic faction in Congress would block black civil rights for the next 90 years.

Southern historians and Confederate memoirists also began to push a kinder, gentler view of the conflict, now known as the Lost Cause school. What had been termed the War of the Rebellion became the Civil War or even the War Between the States. Confederate veterans could be regarded as heroes, and Union and Confederate veterans by the late 1880's were sometimes to be seen standing together in attendance in battlefield celebrations.

So, the short answer is: they stopped fighting because they didn't need to fight. In some ways, they'd won.

Eric Foner: Reconstruction

Wm. S. McFeeley: Grant