For the purposes of this answer, I will assume that you are asking about the formal enfranchisement of African Americans after the Civil War, not about the efforts in the 1960s to ensure that black voting rights were actually respected. If this is the case, there were three main groups, sometimes overlapping, that supported the idea of black enfranchisement.
The first of these groups was, of course, the African Americans themselves. They had a clear interest in being given political power. A second group was a small set of white reformers. These were the sorts of people who, for example, joined the Freedmen’s Bureau. They tended to be motivated by idealism just as the abolitionists had been. Finally, there were the Radical Republicans. Some of this group would have overlapped with the second group. However, the majority of the Radical Republicans were motivated more by politics. They wanted to completely remake the South in the aftermath of the Civil War. To do so, they needed to be in control of the governments of the South. They felt that enfranchising African Americans would be a good way to gain votes for their party. This is not to say that they did not have any idealistic reasons for enfranchising blacks, but there was a large element of political interest in their attitudes.
These were the three groups most in favor of enfranchising African Americans after the Civil War.