There is one unique difference between human brains and the brains of all other animals in this small corner of the galaxy: a capability for conceptualizing the non-existent. Using this very particular skill our species has developed culture, civilization and essentially everything that it means to be human.

But, how did this happen? How did we make the jump from having brains that could solve puzzles to brains that could imagine puzzles for solving?
Obviously the first place to look is the human brain. It is visibly obvious that the brain of a human is larger, especially in the front, where the most critical thinking takes place, than the brains of our nearest primate cousins and our evolutionary ancestors. In the time since the shared ancestor of humans and chimpanzees went extinct, the ancestors of modern humans, the hominids, have evolved with a steady rate of brain growth, each time one species making way for another with a considerable mental advantage. Right up to the end when physically modern humans outlasted and outlived the entrenched Neanderthals of Europe, it was communication which drove the growth of the brain.
Those humans which eventually took Europe from the Neanderthals did so by developing a complex and binding cultural identity, primarily through early religion, which put up a united front against a fragmented Neanderthal population. Language, the ability to communicate in order to form large, more complex and adaptable societies, made the process of natural selection consistent and predictably in favor of big brains plus one other physical feature: sharp tongues.

Anthropomorphism is all good fun, until somebody actually thinks that an animal with a human brain would be able to talk. If you switched the brains of a dog and its owner neither of them would be able to say a word. The dog brain in a human body wouldn't think in words, let alone know how to annunciate them, while the human brain in the dog's body would feel like a recently mute person, totally incapable of phonically conveying the concepts it has envisioned. The human brain will no doubt soon realize that it takes more than a brain to be able to talk. The human brain requires an exceptional set of tools to get across to other brains some of the thousands of ideas that flow through it every hour.
But the brain didn't ask for these tools. The tools came first. The nasal bark of the dog or meow of the cat severally limits the range of communication these animals are capable of, but don't fret, they're not capable of having very complex thoughts anyhow. Without the tools for speaking, cats and dogs have never been able to develop brains big enough to form words and ideas, because, like all things evolution, the two occur in tandem. Even today as we speak, our brains get ever so slightly larger and our mouths (or abilities to use a computer) become ever so slightly better at getting all that large brain info out.
Making the transition from being animals that used our mouths primarily for talking instead of fighting, killing, and eating, had nothing to do with communication at all. Instead, like most great strides of evolution, it was an environmental change that became the catalyst to begin the human journey. A large forest, saturating the plains of Eastern Africa for millions of years resulted in a boon of tree living primates which had first evolved in the earlier, more sparse forests of Africa. There, in that epoch, a great variety of primate species evolved, filling the massive forests to their brim. Eventually, like all good things, the party came to an end and the forest returned to its standard size.

Many of the new primate species just as quickly went extinct, quickly gobbled up by predators while roaming the plains for new habitats. Others established dominance over the forests and eventually evolved into modern apes. The ancestors of humans, though, managed to largely bid adieu to the forest while still surviving. These new primates, the earliest hominids, were still adept to climbing trees, but had also learned to use their hands for other things, like pulling insects out of the ground, or cracking open large nuts, or scavenging the carcasses left behind by large predators. This new system worked and the hominid brain was fed well, especially by protein from the marrow and brains of animals left behind by lions, tigers and bears (I know there aren't bears in Africa) that couldn't get to it with their clumsy paws.

Soon a wide variety of primates evolved living outside of the forest and newer species became less adept at climbing trees and more so at using their hands for other, more human things. This shift completed the long transition from the traditionally quadruped mammal to the fully bipedal hominid, an animal that essentially ceased use of its hands for the purposes of locomotion. The gait of the biped became remarkably different than that of the quadruped. Rather then continue to hunch over, as its primate cousins had done for millennia, the hominids began to stand up straight, taking a big step towards developing a speaking brain.
The hair on top of your head is the remnant of nearly an entire body's worth of long, thick, hair sported by the first hominids. Like their forest dwelling cousins they required the fur to protect from the overbearing African sun when the cover of trees wouldn't do the job. It only seems logical then that those primates which left the forests to roam the plains would become much more hairier, after being exposed to that much more sun. Yet, instead the opposite happens, hominids become less and less hairier than their cousins (even though modern humans of all types have just about the same number of hair follicles as chimpanzees) because they unwittingly begin to use another tool for fighting exposure to the sun: they stand up.

If you take your dog for a walk on a nude beach in the blazing St. Maarten sun, at the end of the walk your naked body will have been exposed to 30% of the amount of sun to which your dog was exposed. Don't worry, you're dog will be fine because of two systems it has evolved to fight all that exposure. The first, of course, is a body full of dark fur that absorbs most of the light's rays, leaving the dog's epidermis relatively safe. The second is all that panting. Dogs, like almost all mammals, don't sweat to cool down. Instead they use their mouth to create a sort of natural radiator, regulating the temperature of the blood right at that critical point before it heads to the brain to perform its most important work.
In order for this radiator to be effective, the animal must have a sizable snout, a protruding nose that can hold a reserve of cold water, blood and mucus (slightly colder than your dog's nose) which brings down the temperature of all the blood flowing through the carotid rete, the pathway through the back of your mouth for blood headed towards the brain. Humans, who are even stingier than radiator-bearing mammals about the temperature of the blood en route to the brain, sweat, in order to maintain our blood temperature system-wide. Not only does sweating make a good alternative to the radiator but it also works much better. Humans can walk exponentially further with our bipedal gait through the hot sun, without a need for water or the shade, than even our primate cousins, let alone other mammals. It took a combination of not only the adaptability of the human brain but also the walkability of the human gait and cooling system to inhabit nearly every corner of the globe.

There the elusive connection between the choice to leave the forest and walk on two legs and the ability to speak is made. It all comes down to sweating, which we can do because we stand upright, and allows us to use our mouths for complex communication rather than as the house for a large radiator. Once hominids developed the physical tools to relay complex ideas it was only a matter of time until their brains became bigger and thoughts became more complex and ideas began to form about things that did not exist, or at least not yet. It is at that moment when the mouth becomes a speaker for the brain that the brain begins to develop all sorts of new and imaginative things to broadcast.