What does it mean to "unfold" a graph? 
edit: more complicated graph
source (problem on pg.21): http://press.princeton.edu/chapters/s7714.pdf
I couldn't find any resources online explaining the unfolding process of a graph in layman's terms and the description from the pdf file doesn't help at all. Take the example from above. I'm confused as to why the numbers from the position they were from changed in the "unfolded graph"
 A: The idea here is that a graph only has the structure of which vertices are connected to which other vertices. One often draws them by assigning a point to each vertex and drawing paths between all pairs of vertices connected by an edge.
What you have here is two drawing of the same graph. One is very tangled and the other is very neat. For instance, take a look at the vertex labelled $2$. On both images, there is an edge between this and $1$ and $3$ and $11$. However, the relative positions of $1$ and $3$ and $11$ are chosen differently so that there are no crossings in the second graph. This is the important property that links the two graphs, and exactly says that they depict the same graph, just with different embeddings.
If you want a physical model which isn't quite right (since strings can't pass through each other), imagine that you run a string between every pair of squares connected by the move of a knight, and you tie all the strings meeting on a square together. You'd get something that looks like the picture on the left. However, you could imagine then moving these strings around into some more appealing form, like the one on the right. This is basically what's going on here.
