For black southerners, the answer to this is quite clear. Blacks in the South would have supported Reconstruction because it helped them. During Reconstruction, governments were actually doing things to help the freed slaves. Black people had rights during this time and many were even part of the state governments.
White Southerners who supported Reconstruction were derided as "scalawags" and were said to be traitors to the South who were simply in it for the money. This view is no longer accepted by most historians. Most historians now emphasize that the "scalawags" were people who supported the Republican Party on principle. Some of them were people from mountain areas who were generally opposed to secession in the first place. Others, like James Longstreet (who was a Confederate general) decided that the Republicans' ideas were needed in order to remake the South.