Evolution By Natural Selection
My Edge answer this year has to be evolution by natural selection. Not only does it explain how we all got here and how we are and behave as we do, it can even explain (at least to my fairly critical satisfaction) why many people refuse to accept it and why even more people believe in an all-powerful Deity. But since other Edge respondents are likely to have natural selection as their favorite deep, elegant, and beautiful explanation (it has all three attributes, in addition to wide ranging explanatory power), I'll have to hone in on one particular instance: the explanation of how humans acquired language—by which I mean grammatical structure.
There is evidence to suggest that our ancestors developed effective means to communicate using verbal utterances starting at least 3 million years ago. But grammar is much more recent, perhaps as recent as 75,000 years ago. How did grammar arise?
Anyone who has traveled abroad knows that to communicate basic needs, desires, and intentions to people in your vicinity, concerning objects within sight, a few referring words, together with gestures, suffice. The only grammar required is to occasionally juxtapose two words—"Me Tarzan, you Jane" being the information- (and innuendo-) rich, classic example from Hollywood. Anthropologists refer to such a simple, word-pairing, communicative system as protolanguage.
But to communicate about things not in the immediate here-and-now, you need more. Significantly, effectively planning future joint activities needs pretty well all of grammatical structure, particularly if the planning involves more than two people, with even more demands put upon the grammar if the planned action requires coordination between different groups not all present at the same place or time.
Given the degree to which human survival depends on our ability to plan and coordinate our actions—and to collectively debrief after things go wrong so we avoid repeating our mistakes (though at a national level we seem really bad at doing that)—it is clear that grammatical structure is hugely important to Homo sapiens. Indeed, many argue that is our defining characteristic. But communication, while arguably the killer app for grammar, clearly cannot be what put it into the gene pool in the first place, and for a very simple reason. Since grammar is required in order for verbal utterances to convey more complex ideas than is possible with protolanguage, it only comes into play when the brain can form such complex ideas. These considerations lead to what I think is accepted (though not without opposition) as the "Standard Explanation" of language acquisition.
In highly simplified terms, the Standard Explanation runs like this.
1. Brains (or the organs that became brains) first evolved to associate motor responses to sensory input stimuli.
2. In some creatures, brains became more complex, performing a mediating role between input stimuli and motor responses.
3. In some of those creatures, the brain became able to over-ride what was previously an automatic stimulus-response sequence.
4. In Homo sapiens, and to a lesser extent in other species, the brain acquired the ability to function "off-line", effectively running simulations of actions without the need for sensory input stimuli and without generating output responses.
Stage 4 is when the brain acquires grammar. What we call grammatical structure is in fact a descriptive/communicative manifestation of a mental structure for modeling the world.
As a mathematician, what I like about this explanation is that it also tells us where the brain got its capacity for mathematical thinking. Namely, mathematical thinking is essentially another manifestation of the brain's simulation capacity, but in quantitative/relational/logical terms rather than descriptive/communicative.
As is usually the case with natural selection arguments, it takes considerable work to flesh out the details of these simplistic explanations, and some days I am less convinced than others about some aspects, but overall they strike me as about right. In particular, the mathematical story explains why doing mathematics carries with it an overpowering Platonistic sense of reasoning not about abstractions but real objects—at least "real" in a Platonic realm. At which point, the lifelong mathematics educator in me says I should leave the proof of that corollary as an exercise for the reader—so I shall.