When the South Succeeded from the Union, they had left the Union completely.
The person cutting the branch, or the southern states, off of the tree, or the Union, represents the South succeeding from the Union.
So this cartoon would fit into the narrative about Southern Succession
The southerners would feel that this is a good representation about what happened, but the northerners would feel that the southerners had never truly succeeded in the beginning, until later.