According to the body of science, we still have work to do before the vast majority of evolutionary scientists can answer yes. Some say "yes," some say "no," some say "maybe in the future," some aren't sure, as the Templeton Big Question Series indicates. In the end, it really depends on what you mean by "evolution," "explain," and "human nature." (Oh boy, here we go!)
First, let's start with the easiest word and define "explain" as "to adequately describe the basis for a new trait, why and how it spreads, and how it grows in complexity." That definition is easier than the other two words, but it already shows potential difficulty. It's not just how a trait appears but how it changes and how it spreads.
Secondly, the word "nature." Nature is the collective set of traits belonging to a species. Obviously evolution can explain some aspects of human nature, like why we have two legs instead of six. But the title question is implying whether evolution can explain aspects of human nature that appear to be special and unique beyond animal nature -- humanity's moral nature and behavior, cooperation, reciprocal altruism, and religion, as well as the appreciation of beauty, the desire to make and perform music, consciousness, and planning into the future.
Third, the word "evolution." If we only mean genetic evolution when we say "evolution," I will clearly answer the title question no; genetic evolution does not fully explain human nature as described above. Professor Joan Roughgarden illuminates quite well that some human traits are formed before any life experience and are properties of the individual like skin color or the ability to taste bitter flavors. However, some traits are formed and learned through life experience and social interaction during development as one matures. With advanced language skills, through teaching, imitation, and learning, cultural traits can be transmitted at a very fast rate, as Professor Martin Nowak notes. Cultural evolution and genetic evolution can even interact with each other when those with certain cultural traits are favored for genetic reproduction, creating a positive feedback loop, as Professor Eva Jablonka describes.
Now that we've answered the question regarding purely genetic evolution, let's broaden the question: Does socio-genetic evolution explain human nature? Professor Frans de Waal says it does and has noted natural, human traits in other animals. He has observed chimpanzees living in highly structured groups utilizing rules, mutual cooperation and inhibitions. He has seen chimpanzees mutually embrace, console a victim chimpanzee, open a door for another chimpanzee, and refuse to work when being fed cucumbers while another is fed grapes. Agreeing with Professor de Waal, Professor Lynn Margolis argued that human forms of togetherness, compassion, religion, and planning can be explained through evolution, while Professor Geoffrey Miller purports that evolution can even explain art. Still, there are disagreements not just between supporters and opponents of evolutionary theory but between evolutionary scientists themselves. Author Robert Wright believes that such disagreements between evolutionary scientists imply that there are a plethora of evolutionary theories for human uniqueness rather than a dearth. To conclude, there still is not a consensus on a theory that fully explains certain parts of human nature like language or consciousness, as Professor Simon Conway Morris elucidates. Remember, our definition of "explain" is to describe how a trait first appears, as well as how it spreads and how it becomes more sophisticated. Currently, there is no adequate, agreed-upon theory that addresses this for language, consciousness, or even love.
Perhaps at the heart of this question are really two deeper questions. First, are humans special and unique, above and beyond other animals? Second, is the specialness or unique nature due to a God? In other words, is there a God?
The historical debate over this question follows a pattern. Someone says humans are special because they use tools; then scientists show how animals use more primitive tools. Someone says humans are special because they have language; then scientists show how other animals have more primitive forms of language. And so on and so on. Tools, language, art, non-procreative sexual intercourse -- the list goes on.
But does finding tools or language in other animals take away the uniqueness of humanity? To answer this, Karen Armstrong helps highlight a good example trait: religion. She said the one thing that sets humans apart from animals is our capacity for religion. Is this uniqueness true? In actuality, yes. Language is another example. Even if other animals have symbolic forms, whether for language or thought, ours is clearly more developed and thereby unique in that development. The question is not whether a trait is different or special in development or advancement; the question is the cause of this specialness: is it the result of evolution or something else (God)? And that is very hard to ascertain.
Professor Jablonka notes that finding the source of such uniqueness is very difficult because it occurs at the confluence of emotional, social, and genetic foundations. There's still much more work to do, but she feels confident the framework is in place for explaining the origins and evolution of symbolic systems, at the least. For me, as a non-expert in evolutionary science who has been reading the wide views on the subjects from evolutionary scientists, the jury is still out on this debatable issue.
Regardless of your viewpoint, let's imagine that we have proven that evolution does explain all aspects of human nature -- altruism, reciprocity, religion. How is that answer related to God? The original question serves as a proxy question about the existence of God, yet, in reality, it doesn't. It's not related. As Dr. Francis Collins clarifies that even if evolution can fully explain altruism, it does not exclude the existence of God.
I guess it might be the reverse of the god-of-the-gaps theories. Just like the presence of something special without current explanation does not prove God's existence (contrary to god-of-the-gaps theories), the presence of something special with a fully natural explanation does not prove God's nonexistence.
In some sense, the answer to the question still leaves more questions. Professor Jeffrey Schloss and Robert Wright agree at least on this: Even if evolution adequately explains falling in love, sentience, and beauty, it still wouldn't tell us what those things are or what they mean. With the never-ending aspects of human nature to explain, I sometimes wonder about how much effort we give to this question and how we prioritize this search with other demands upon the world. I believe Professor Roughgarden said it best. There's nothing wrong with asking how we evolved different parts of human nature. However, some aspects of this endeavor are more practical than others and more likely to succeed.