My example of a deep, elegant, beautiful explanation in science is John Maynard Smith's concept of an evolutionarily stable strategy (ESS). Not only does this wonderfully straightforward idea explain a whole host of biological phenomena, but it also provides a useful heuristic tool to test the plausibility of various types of claims in evolutionary biology—allowing us, for example, to quickly dismiss group-selectionist misconceptions, such as the idea that altruistic acts by individuals can be explained by the benefits that accrue to the species as a whole. Indeed, the idea is so powerful that it explains things I didn't even realize needed explaining until I was given the explanation.
I will now present one such explanation below to illustrate the power of ESS. I should note that while Smith developed ESS using the mathematics of game theory (along with collaborators G. R. Price and G. A. Parker), I will attempt to explain the main idea using almost no math.
Think of common animal species like cats, or dogs, or humans, or golden eagles. Why do all of them have (nearly) equal numbers of males and females? Why are there not sometimes 30 percent males in a species and 70 percent females? Or the other way around? Or some other ratio altogether? Why are sex ratios almost exactly fifty-fifty? I, at least, never entertained the question until I read the elegant answer.
Let's consider walruses: They exist in the normal fifty-fifty sex ratio, but most walrus males will die virgins, whereas almost all females will mate. Only a few dominant walrus males monopolize most of the females. So what's the point of having all those extra males around? They take up food and resources, but in the only thing that matters to evolution they are useless, because they do not reproduce. From a species point of view, it would be better and more efficient if only a small proportion of walruses were males, and the rest were females; such a species of walrus would make much more efficient use of its resources and, according to the logic of group-selectionists, would soon wipe out the actual existing species of walrus with the inefficient fifty-fifty gender ratio. So why hasn't that happened?
Here's why: because a population of walruses (you can substitute any of the other animals I have mentioned, including humans, for the walruses in this example) with, say, 10 percent males and 90 percent females (or any other non–fifty-fifty ratio) would not be stable over a large number of generations. Why not? In the 10 percent males and 90 percent females of this example, each male is producing about nine times as many children as any female—by successfully mating with, on average, close to nine females. Imagine such a population. If you were a male in this kind of population, it would be to your evolutionary advantage to produce more sons than daughters, because each son could be expected to produce roughly nine times as many offspring as any of your daughters. Let me run through some numbers to make this clearer: Suppose that the average male walrus fathers ninety children, only nine of which will be males and eighty-one females, on average, and the average female walrus bears ten baby walruses, only one of which will be a male and nine will be females. Okay?
Here's the crux of the matter: Suppose a mutation arose in one of the male walruses—as well it might over a large number of generations—that gave this particular male walrus more Y (male-producing) sperm than X (female-producing) sperm. This gene would spread like wildfire through the described population. Within a few generations, more and more male walruses would have the gene that makes them have more male than female offspring, and soon you would get to the fifty-fifty ratio that we see in the real world.
The same argument applies for females: Any mutation in a female that caused her to produce more male than female offspring (though sex is determined by the sperm, not the egg, there are other mechanisms the female might employ to affect the sex ratio) would spread quickly in this population, bringing the ratio from closer to fifty-fifty with each subsequent generation. In fact, any significant deviation from the fifty-fifty gender ratio will, for this reason, be evolutionarily unstable and through random mutation soon revert to it. And this is just one example of the deep, elegant, and beautiful explanatory power of ESS.