If I read the code correctly the meanings are as follows:

parents[v] : The vertex that comes right before v on a shortest path from the source to v.

dists[v]: The length of the shortest path from the source vertices to v.

preds[v]: The vertex that comes right before v on some shortest path from the source to v. The difference to parents is, that parents considers only one shortest path, but there could also be multiple. Is only created when allpaths is true.

pathcounts[v]: The number of shortest paths from the sources to v.

closest_vertices: A list of all the vertices of g in order of shortest distance to the sources. Is only created when trackvertices is true.