First of all, it is not really accurate to say that sectional hostilities abated quickly. Yes, the two sides never came anywhere close to war again. However, there continued to be hard feelings, particularly on the side of the South.
The North stopped feeling anger towards the South after the war because there were no longer any serious issues dividing the two sections as sections. The South no longer had slaves and so the North no longer felt threatened by slavery. The South was treating African Americans poorly, but that did not matter to most in the North.
The South continued to feel anger, but it was not really expressed in any political way. They were upset at having lost their way of life, but that did not translate into (once Reconstruction was over) any real political differences with the North that would keep the anger alive.
The previous post by pohnpei was pretty accurate concerning the North's attitude of just ignoring the South after the conclusion of the Civil War hostilities. Southern hatred of the North certainly DID NOT abate quickly; more accurately, it gradually diminished following the deaths of generations of Southerners, leaving only a New South populated by men and women who, like Northerners after the war, just didn't care anymore.
The South was beaten long before Lee's surrender at Appomattox: Its towns and crops were destroyed, its slaves were freed, its men were dead and maimed. They were not likely to rise again, but Southerners would not forget quickly. In response to the influx of carpetbaggers and scalawags, dishonest Republican politics, and former slaves with rights now equal to their former masters, the South fought back with the rise of the KKK, Democratic politics and, eventually, Jim Crow laws. Civil War veterans groups, though found in the North, flourished in the South, their influence only fading as the last veterans died away. Today, Civil War reenactments are annually held on most of the famed battlefields of the South, as well as on many of the minor sites of obscure battles--another sign that remembrances of the war are not yet dead in the South.