African Americans gained political power during this time in a number of ways. Let us examine two of the most important of these ways.
First, this was a time in which more "majority-minority" districts were being created. Courts ruled that it was proper (and in some cases required) to create legislative districts in which African Americans were a majority. This more or less guaranteed that more African Americans would be elected to Congress.
Second, this was a time when the African American vote was becoming much more important for the Democratic Party. Many blue collar white voters had left the party in the Nixon and Reagan years. That left African Americans as a vital constituency for the Democrats. This fact gave them more political power as well.