A lot of fluff around the central point which doesn't seem incredibly impressive...
First, an organism (or a bunch of organisms, a population) changes its functional form — its phenotype — by making broad changes in gene expression. Second, a gene emerges that happens to help lock in that change in phenotype. Third, the gene spreads through the population.
How is he defining "gene"? The coding region of the sequence that directly leads to the final protein product? The sequences for transcription factors and RNAi pathways that define when to activate or inhibit that other sequence? The sequences for the various methyl and acetyltransferases that increase or decrease the accessibility of that sequence for transcription? Other bits and pieces that we just don't know that much about yet?
You start running down these critters. As you do, certain genes ramp up expression to build more muscle and fire the muscles more quickly. You get faster. You’re becoming a different animal. You mate with another fast hunter, and your kids, hunting with you from early on, soon run faster than you ever did.
How does you change its own gene expression? Do you rely on epigenetic modifications to DNA / histones? Do learn to change your behavior and train your kids on that? These flexible aspects of your genome are usually encoded in some source gene one way or another, down to those that support sexual over asexual reproduction.
And on top that, the example with mate matching selects for the genes in the population that just produced the better hunters in the new context. And speed isn't exactly a one-gene phenotype, it's going to be affected by spots all over the genome, some of which work together in combination, of which don't. Some of which encode the muscle fibers, some of which control those other gene expression.
As an aside, I think a much more better illustration of the concept of a "selfish gene," are transposons, which repeatedly replicate (usually-useless) sequences all over the genome and are carried on, like flies on a beast's back, from generation to generation as long as they don't cause trouble.