I would also add the revenge aspect of Reconstruction legislation. Many Northerners wanted to punish the South before allowing them back into the Union. There was quite a bit of debate/arguing over the ease vs. difficulty states should face in being restored their place in the U.S.
The amendments became a part of the punishment, but also the oaths taken by former soldiers would play a part in that as well, as well as legal action taken against political leaders, allowing former politicians back into office, etc.
There are a lot of different laws that governed Reconstruction, so it is not really clear which one you mean. I will try to give you an idea about the major aspects of Reconstruction in general.
The most important laws pertaining to it were
- The "Civil War Amendments." These were the 13th, 14th, and 15th Amendements. They made slavery illegal, made blacks citizens with equal rights, and gave them the right to vote.
- The breaking up of the South into military districts. This took away the Southern states' rights to democratically elect their own leaders. In place of that, it instituted military governments that would rule the states without their consent.