Question

Radical Republicans stated that Southern states had to in order to be readmitted into the Union.

Answer 1

Because they believe in a strict interpretation of the Constitution, meaning they relate everything back to that declaration, and in that it takes about "one nation". If they were to give free reigns to the Confederate states, they would slip under their radar, plan how to secede, and form their own independent country. The radical Republicans did not want that.

Another reason would be because they wanted the Confederacy to pay the repercussions for the Civil War. They wanted total power over them to show that THEY are the leaders of the country and they will not be made fools of. They had to establish authority in order to gain back respect and tolerance from the confederate states.