Question

Why only man has highly developed brain? Why other Animals don't have that much developed brain?

Answer 1

It's misleading to say that the human brain evolved "more".  Every brain is specialized to its own niche.  We declare the human brain to be best because that's the one we have.  Your brain can't do some of the feats that the brains of other creatures can.

Still, even setting that aside, it's clear that the human brain has been successful in a way that no other creature has in the history of the planet.  Neurologists and philosophers debate exactly how to characterize the difference.  Many focus on its capacity to process abstractions and symbols, but it's still not 100% clear what makes the human brain different from that of its closest relatives.

Whatever it is, it's been very successful.  The human animal exists in practically every climate on land and makes inroads into much of the water, too.  The species is so successful that it is wiping out many other species and is altering the face of the planet so much that we may in a new "Anthropocene" era.

The human brain isn't necessarily any "more" evolved, but it did find a remarkable niche that nobody else had before, and exploited it like crazy.  We see echoes of its specialization in other animals, from chimpanzees to birds.

The human species pays a price for it.  The brain consumes a considerable fraction of the body's energy.  Squirrels do perfectly fine with much smaller brains, allowing them to live on the much less dense sources of energy that they use.

The human brain appears to have been in the right place at the right time.  Human-like brains may only be possible among warm-blooded creatures, since the brains of cold-blooded creatures may not be able to keep up the energy required for complex thought.  That would make a big energy-requiring brain less useful.
 A lot of different factors all converged: upright motion (freeing the hands for deft manipulation) and a freely-moving vocal apparatus (allowing complex language) being two prominent ones.  

Humans likely evolved as the climate was changing from forest to grassland.  They became runners, freeing up their hands to use tools.  They used this running to hunt, providing a lot more energy than vegetables can provide.  They used the communications to hunt collectively, a much more effective technique than hunting individually.

This happened over the course of a few tens of thousands of years, about 100,000 years ago.  The distance means that it's hard to tell which occurred first and how the various factors worked together to create the human organism as we see it.