Can the theory of evolution explain how your brain evolves to become unusually outstanding compared to other animals and why it only occurs in homo sapiens?

Yes it can. Evolution is about adaptation. So if a life form needs intellective to survive, then so be it.

We humans started out like apes, but slowly began to get intelligence. How? Well it did not happen overnight, and that is one thing creationists have a hard time understanding, but with an open mind one can see it is possible.

We know that thousands of years ago (hundreds of thousands) we were hunter-gathers, people who traveled from place to place looking for shelter and food, like animals, but we suddenly started settling down, made our homes out of the materials around is, and domesticated animals, why did we do this?

First we began to recognize patterns in our environment, this was accomplished with help from better memory with our bigger brains, so we saw things like how seasons effect plants, and the food they provide, and how we can use this to our advantage.

Any animal can do this, even the Octopus (believe it or not) has amazing reasoning and memory skills, as tests have shown, which it evolved (not as advanced as us by far, but still the beginning of intelligence as we know it).

We also saw that the things around us can help us do things, like a stick can be good for defense, and that by seeing patters, we soon learned that sharper sticks were more effective. We saw that rocks are hot when rubbed together, and how that makes sparks that catch wood on fire. We saw that the skins of animals are warm, so we use those in the winter, etc, etc, etc.

We also worked in groups, relying on each other for help and guidance. Our parents learn from our grandparents, and we learn from our parents, and we teach our children. Like animals, we stayed in groups for protection, unlike animals, we passed on our skills and knowledge of experience, and that is how we have made our progress.

That is how we adapted, so intelligence is part of our evolution.

The reason why it happens only in us is because it is SO RARE. But we see different levels if intelligence in other animals around us, it is just rare for intelligence to advance this far.