Question

How and why protons and electrons are positively and negatively charged??

Answer 1

No one truly knows. There is no testable theory that explains why this should be. Theories of everything (such as superstring theory or loop quantum gravity) ultimately wish to be able to derive these from first principles, but we are a long way away from even coming close. Until that time, they are just one of those things.

I dont know much about loop quantum gravity, but in superstring theory the idea is that teh properties of each particle is defined by how a one dimensional string vibrates through 11 dimensions of space time. Charge would therefore be one form of vibrational mode, mass another etc. The problem with string theory though (along with other theories of everything) are that they are lacking in the ability to test, and until they are tested for falsification, they are nothing more than scribbles on a piece of paper (or codes in a super computer), regardless of how elegant they may be.

Answer 2

First, it's a historical choice to define electrons as negative charges, so protons became positive one. What is important, is the interactions on each other. If you want more information on the material nature, see around quark, but you touch a sensitive point of lastest theory. Why, we don't really know.