How does node expansion work in a graph for AI search?
It seems to me that you are confusing the graph itself with the search tree produced by some search procedure.
As you state, you can't go from one node to the next unless that node knows its neighbors (and not only its own parent). But the search tree is (perhaps implicitly) generated from a search procedure you perform on the actual graph. Perhaps this is what the authors mean with the description you quote.
The actual graph on which you do a search in you can represent via an adjacency list. There, given a node, you can locate its neighbors and the search progresses using that information.