Most of your essay is very clear and strong, but your introduction could do with some work.

What would happen if you removed the passive in the intro?

It is accepted that wealthy nations should help poor countries by helping with food, education, and financial aids. Nowadays, you can see a lot of organizations providing this kind of aids for poor countries. However, government of wealthy countries and their people take a responsibility to help poor people in Africa, Asia, and other part of the world. There are more reasons into why we should help this matter. In this article I explain that this it is the responsibility of both wealthy countries and poor countries to look after their people.

 The introduction goes around the topic. One sentence ("There are more reasons into why we should help this matter") adds very little.  The word 'however' is a little strange, given your first sentence. I think it would be clearer if the whole paragraph was shorter: 

Wealthy nations and donor organizations help poor countries in Africa, Asia, and other parts of the world with food, education, and financial aid.  However, I don't believe that only rich countries have this duty. In this essay I explain that it is the responsibility of both wealthy countries and poor countries to look after their people.

 It would be clearer and stronger. Just start with a good situation introduction (explain or describe what the situation is) and your thesis sentence (what you are going to do).

I like your ideas and how they are developed in the rest of your essay. The only change I would make it to remove another unidentified passive: 

It has been aptly believed that teaching a man to fish is better than giving the daily fish to them.

Perhaps you could rewrite this more simply as:

Teaching a man to fish is better than giving him a fish.

Thanks for your essay and I hope you can help some other writers here! 

First of all, we are all living together in one world so we cannot be carless about the situation of people in poor countries. We are all human beings. We cannot close our eyes to the famine, orphan people or starveling country.

Second, it is fact that if we help poorer nations it will protect our nations and economy from any future war or massive migration from those parts. Moreover, aiding the poor countries can help our industry to sell them their products and services they need. In one word, helping them would help our country such as increase our production, wealth, and whole economy.

In addition, wealthy nations cannot take all the responsibility of poor countries and helping them at all times. It is responsibility of their governments and people to make better use of donations from the wealthy nations to develop their country. On the other hand, most developed country should transfer education, funds, and knowledge to them rather than direct help. It has been aptly believed that teaching a man to fish is better than giving the daily fish to them.

From these facts, wealthy countries should continue to help poor countries and trying to help them to restructure their economy by converting from regular help to technological assistance and management support. But it will not be achieved without taking responsibility by both governments of poor countries and developed countries.