How can you be "re-admitted" to something you never left?

The Union government and Abraham Lincoln never recognized the right of the Southern states to secede, and maintained that they never left the Union. Yet after the war was over, all of the former Confederate states had to be "re-admitted" to the Union, a condition of which was ratification of the 13th Amendment. How can this be? How can you be "re-admitted" to something you never left?