Gentle introduction to graph isomorphism for bounded valance graphs
The algorithm for bounded-degree graph isomorphism is so closely tied to (permutation) group theory that I doubt there is an introduction that uses groups "only gently." However, you might consult Paolo Codenotti's Ph.D. thesis for more complete background. He doesn't cover bounded-degree graph isomorphism exactly, but covers the tools needed for it (and the rest of the thesis is about bounded-rank hypergraphs, extending the best known algorithm for general graph isomorphism to the bounded-rank hypergraph case).

You may also find the book Group-Theoretic Algorithms and Graph Isomorphism useful, as it covers most of the background necessary (Chapter 2, "Basic Concepts", is 47 pages) and is a much more leisurely exposition than most of the published papers on the topic.