Visualized definition of cohomology I cannot imagine how cohomology is related to graph theory, actually I read solid definition from wiki, and to be honest, I cannot understand it. 
e.g I know what is homology (in simple term), group of functions such that I can continuously convert each of them to another one, but, is there similar visualization method for cohomology? (I'm not looking for exact definition, I want to imagine it, actually this is in graph theoretic concept). for more information see introduction of this paper. I want to understand it in this paper, how is useful? how to imagine it? 
P.S1: my field is not related to group theory, and as in introduction author wrote, this paper doesn't need deep group theoretic definition! and I don't want to be deep in group theory. Just looking for simple way to understand them.
P.S2: I think I can imagine what is free group (which is in introduction of paper), at least by Calay graph seems to be easy to imagine it. 

I think it's fair to see to say that, in general, cohomology is not used in graph theory, and so your question does not have an answer. 
Schrijver's paper is one of the few exceptions, and he states that he is just using the language. So I think you best advice is read some introductory texts to algebraic topology. I say "some" because the introductory texts tend to treat cohomology or covering spaces/free groups, but not both. Note that covering spaces do turn up regularly in graph theory, see e.g. voltage graphs.
