Understanding graph minor theorem
The following book covers some material related to the proof of the graph minor theorem (Chapter 12). 

Reinhard Diestel: Graph Theory, 4th edition, Graduate Texts in Mathematics 173.

The author states: "[...] we have to be modest: of the actual proof of the minor theorem, this chapter will convey only a very rough impression. However, as with most truly fundamental results, the proof has sparked off the development of methods of quite independent interest and potential." 

An electronic version of the book can be viewed online. 
http://diestel-graph-theory.com/