I am trying to understand the concept of clique graph. So I found this page. But I do not understand the example and what “graph intersection” is.
Can somebody explain to me why $K_4$ is a clique graph of $G$?
All I understand so far is that this particular graph $G$ has 4 cliques they are all $K_3$.
If we have some collection of sets, the intersection graph of the sets is given by representing each set by a vertex and then adding edges between any sets that share an element. (Wikipedia has a nice picture in the intersection graph article.)
The clique graph is the intersection graph of the maximal cliques.
As you note, $G$ has four maximal cliques.
Each of the four cliques shares at least one vertex with each of the other cliques, so in this case the intersection graph is complete.
Suppose you go through and circle all of the maximal cliques of G (which, as you noted, are all G_3s). These will be the vertices of the clique graph. When two (or more) circles overlap at a vertex in the original graph, those vertices will have an edge between them in the clique graph.