How did the unique features of human intelligence evolve? I have been debating the following topic with a friend.


She argues that humans do not descend from chimpanzees or orang-utans, because if we did, such animals would share the same cognitive thinking that humans have.
I argue that we do descend from that such animals, and that a high protein diet has allowed early humans to develop a better memory, intelligence, and other mental processes.


Questions


What is the  the real origin of human intelligence? 
If non-human animals do have intelligence too, why is their intelligence not as advanced as humans?
If non-human animals do have intelligence too, why is their intelligence not as advanced as humans?


Notions like “advanced” or “better” really have no place in evolutionary thinking. Again, evolutionary fitness is about self-reproduction and success compared to whatever competition is present at any moment. There is no force “optimizing” species to meet some external criteria of “progress”.

One important observation is that (relatively) simple species have in fact been quite successful (in terms of number of individuals, total biomass, capacity to survive climate changes and other disruptive events, etc.). Humans have been flourishing very recently (few thousands of years), colonizing big parts of the earth, reaching large population numbers but before that there is evidence that they went through a severe population bottleneck as recently as 70000 years ago. Great apes populations are also not very large and all species are considered endangered. 

Sustaining a large brain and human intelligence is in fact extremely onerous. The human brain requires a lot of energy. Furthermore, it has become too big to go through the mother's womb so that it still undergoes growth after birth, meaning that human babies require a lot of care and protection. All that suggest that “high intelligence” is not always such a big advantage, evolutionary speaking. As often, there are trade-offs and various successful “strategies”.

In any case, you might be interested in Wikipedia's article on the “Evolution of human intelligence”.