One of the goals of Reconstruction (in some people's opinion) was to destroy the Old South and make it more like the North. Part of that process was going to be the creation of a new class of Southern elites who would be involved in government.
The Amnesty Act helped to prevent this from happening. It allowed almost all of the former Confederate elite to regain their right to vote and to hold office. This meant that the political power structure of the South would not really be changed. If it did not change, Reconstruction would be (in that way, at least) a failure.