As promised, the first two thirds of the book was not nearly as compelling as the last few chapters, since it was mostly setting up the ground for the central argument and knocking aside the various misunderstandings and misapprehensions that would impede it. By time the final three chapters came, though, the argument being made, rather than seeming shockingly revolutionary and world-changing, actually felt more inevitable, inescapable, almost obvious. It doesn't take a moment's thought to realize that it wasn't any of those things; that, like many of the moments of brilliance in scientific history (even Darwin's insights) it's obvious in hindsight only. I don't mean to sound like I'm diminishing the book; in fact, it's a compliment that when we got to the idea it was just the end of an ineluctable progression of logic.
By its nature it's hard to sum up briefly in a way that doesn't suggest more misapprehensions than actual understanding. But this is an analogy I came up with that I think isn't too bad.
When you're calculating a spaceship's trajectory from the Earth to the Moon, you have to account for the gravity of the Earth, the Moon, and the Sun, but you generally don't need to think about anything else. Theoretically, it's true that the spaceship itself, Jupiter, Pluto, Proxima Centauri, the Andromeda Galaxy, and an electron in a particle of interstellar dust 11 billion light years away, all exert a gravitational force on the spaceship. That's a real, actual force, not a mere technicality: particles genuinely are affected by it and genuinely move accordingly. But you can ignore these things because of the inverse square law of gravitational propogation: the force drops off very very sharply with distance, far faster than it increases with mass. So while the Andromeda Galaxy masses billions of times more than does the Moon, the distance is so much greater that its effect is almost immeasurably tiny.
Every physicist, and even laypeople with a good scientific grounding, will know both that these gravitational effects are real, and that they can be ignored. And therefore, everyone knows that ignoring them is a convenient shortcut, not a truth. If you're a layperson, even a fairly well educated one, your thinking about the theory is likely to be polluted by the ubiquity of this shortcut; and you're probably likely to imagine that that tendency is endemic, and might also affect the professional physicists. (And it can, though it's far, far less likely to do so than the layperson typically imagines it to be.)
Now let's jump to genetics. Every student of genetics and evolution knows that Mendel's seminal analysis of pea plants, with a gene "for" being tall or short, are a vast oversimplification, but a meaningful one. Obviously, a single gene can't make a plant be tall: it takes all the genes just to make the plant grow in the first place, and it can't be tall if it doesn't grow. Many genes will affect growth, as will many environmental factors. Any given thing you observe about the plant can be seen to be at the center of a web of thousands of causes. But when you're trying to figure out genetics, it is still meaningful to say that this is a gene "for" height, because all other things being equal this gene will make the plant grow taller than its alternative (its "allele"). We can prove this, despite the impossibility of all other things being truly equal, using well-understood statistical methods, in which the "all other things" will even out in aggregate.
That a particular gene makes a plant taller than its allele is called that gene's phenotype (the physical expression of a gene). The game-changing revelation of Dawkins's book is that the phenotype does not have to stop at the edge of an organism's body, but extends out into the world. Like gravity (though with nowhere near the precision of gravity), phenotype diminishes with separation. And like gravity, this diminishment means that for many practical questions of biology and evolution and genetics, you can safely ignore the phenotype effects of far-removed genes. But those effects are there: they are real, and sometimes they're the only explanation for things that otherwise wouldn't make sense, that would seem anomalous.
To make the analogy complete (or as complete as it's going to get), though, you have to imagine that in the world of biogenetic evolution, everyone was essentially a layperson. In a real way, the science of evolutionary genetics is so much less advanced than physics, that the situation I described earlier (where laypeople are more likely to make mistakes in theoretical thinking due to their shortcuts) applies even to the best experts in the field. (Just as, for instance, Aristotle made similar mistakes in thinking about physics when that science was in its infancy.)
Much of the most important thinking about evolutionary genetics in the years since Darwin has been skewed by the idea that a phenotype ends at the borders of an organism's body, in much the same way that it has been skewed by the idea that genes serve organisms and not the other way around (the mistake Dawkins corrected in his earlier book). Dawkins is simply nudging the science along one step closer to leaving its infancy. That he's been able to make two such game-changing observations in a single career is pretty extraordinary. One could make the following analogy: Newton is to Einstein as Darwin is to Dawkins.
If you're curious about the actual extended phenotype itself (the ways that the phenotype extends beyond the body, and why the body's border is almost as arbitrary as the borders of the nucleus, the cell, or even the messenger RNA), I recommend reading the book!