Civil War reenactment has a long history. Early reenactments consisted of veterans from both the North and the South fighting mock battles against each other around the turn of the 20th century. Later, reenactment had a surge of popularity around the centennial of the Civil War, in the early 1960s, and then another boost in popularity in the 1980s and 1990s, around the 125th Anniversary of the war.
The Battle of the Crater, fought in 1864, had a reenactment in 1903. One of the traits about reenactments generally--and which this 1903 reenactment emphasizes well--is that it glosses over the brutal aspects of war, as well as marginalized groups which fought in the war. The Crater, a small battle in the Petersburg campaign, was notable for two things--an innovative use of mining technology to blow up a Confederate fortification, and the brutal massacre of African-American USCT soldiers fighting in the battle. Both of these events were, by all eyewitness accounts (at least from the people who weren't supporting the massacre--of which there were many), absolutely horrifying to witness, and neither was shown in the 1903 reenactment. Nor were there any USCT veterans present at the reenactment, nor any mention of them being killed as they attempted to surrender.
The early reenactments--at the Crater and elsewhere, such as at Gettysburg in 1913, on the fiftieth anniversary--had the aims of fostering nation unity between white Americans who had fought on both sides during the war. In his speech at Gettysburg on July 4, 1913, then President Woodrow Wilson said "We have found one another again as brothers and comrades in arms, enemies no longer, generous friends rather, our battles long past, the quarrel forgotten except that we shall not forget the splendid valor, the manly devotion of the men then arrayed against one another, now grasping hands and smiling into each other's eyes." (You can read the full speech here if you're interested). Clearly Wilson is advocating for a memory of the battle which does not dwell on the historical fact of the battle outside of the abstract memory of the participants bravery.
In general, as we approach the modern day, participation in Civil War reenactments seems closely correlated with notable anniversaries of Civil War battles, which generally drive interest in all areas of the Civil War. Reenactments regained some popularity around the centennial in the 1960s, and were most popular in the 1990s, with the Gettysburg reenactment of 1998 thought to be the largest reenactment to date, around the 125th and 135th anniversaries of the war. Obviously, every person involved in reenacting has their personal reasons and motivations for doing it, but many seem to enjoy the sense of community with other reenactors, spending a weekend outside with their friends, or the connection they feel with the soldiers or their ancestors.
Many are also deeply critical of reenactment as well, in part for the reasons I mentioned above--war is brutal, and reenactments are necessarily not so. Some reenactors (a small minority) strive endlessly for historical authenticity, starving themselves and going for marches barefoot for twenty miles carrying a full pack, though this is obviously not particularly appealing to a wider audience. (Some of these actually refuse to take place in the mock battles, because unless they use real bullets, they are sacrificing historical authenticity). Reenactments, such as at the Crater, nearly always gloss over the bloodier aspects of the battle, and many critics say that reenactments make a game of something that should not be taken lightly.
In summary, reenactments have a long history dating back to close after the Civil War, and were used as attempts to foster national unity between White Americans. Later, reenactments become more popular around significant anniversaries of the Civil War, specifically peaking in popularity in the 1990s, after the 125th anniversary from 1986-1991.
Bibliography: Levin, Kevin M. (2012) Remembering the Battle of the Crater: War as Murder
Horwitz, Tony. (1998) Confederates in the Attic: Dispatches from the Unfinished Civil War.