Finding k shortest Paths with Eppstein's Algorithm
It's been long enough since I wrote that, that by now my interpretation of what's in there is probably not much more informed than any other reader's. Nevertheless:

I believe that the description you're looking for is the last paragraph of the proof of Lemma 5. Basically, some of the edges in P(G) (the "cross edges") correspond to sidetracks in G (that is, edges that diverge from the shortest path tree). The path in G is formed by following the shortest path tree to the starting vertex of the first sidetrack, following the sidetrack edge itself, following the shortest path tree again to the starting vertex of the next sidetrack, etc.