We don't know. There are no well constructed theories with a solid evidential backing. There isn't even anything close.
My fellow posters have put forward the major contenders, but the truth is that we just plain don't know. Partly this is inevitable; it's difficult to imagine where we'd even get compelling evidence from given the lack of information that the fossil record can provide on the subject. Partly it's that our knowledge of how the brain works, and the molecular evolution of the human brain, are extremely rudimentary.
Even the brain power of a goldfish is astounding a few cells of brain matter and you can actually train a goldfish.
The brain of a goldfish does not contain a "few cells", unless you consider 7 and a half million "a few".
My claim is that our kind of conscious intelligence has only evolved once. I am not claiming that other animals don't display some degree of intelligence, they obviously do. But it's equally obvious that their intelligence is not of the same order as ours. Not even close.
I'm puzzled by your inclusion of "conscious" here?
I agree with you that human-level intelligence has only evolved once in the history of life on Earth. But I'm not sure how conscious comes into it? Are you suggesting we're the only conscious species? Or that other species have the same intelligence but are not conscious?
Just thousands, to be precise the last two thousand.
We know for a cast iron fact that this is false. Human populations show the signs of natural selection for resistance to various diseases.
The population increase that humans went through in the last 2-3 thousand years has few parallels in nature, and there is plenty of evidence showing that selection does not operate in expanding populations, I can dig up refs if you want. But to put it simply, natural selection operates under the premise that some of your progeny will die (and with it presumably your "weak" genes), so in order for natural selection to operate, a certain percentage of the population has to perish. But in today's society this is not happening anymore, or in other words, both the fittest and the weakest are surviving. If both are surviving there is no selection.
I would be interested in your refs because the description you have given makes no sense. All selection needs to act is inheritable differences in relative fitness - a gene that produces more living offspring can be selected for in an expanding population as easily as one that produces a drop in living offspring in a reducing population. It would only make no difference if both the fittest and the weakest survive unless they both also go on to have exactly the same number of surviving offspring themselves - at which point they have the same fitness anyway.
It's also simply not true that everyone was surviving for last two thousand years, throughout most of that period the majority of people born never lived long enough to have children of their own.