You have a solid start to your essay here in that you have identified several important elements of your argument.
In your introductory section, you should discuss what you believe to be the factors responsible for advances in science. Your list might include a well-developed, flexible educational system which simultaneously funds researchers and leaves them free to follow their own lines of inquiry. A second factor is contact with other cultures, so that researchers do not need to reinvent the wheel as it were but can build on other people's innovation. A third factor is a society that views intellectual inquiry in a positive fashion.
Your second, or transition paragraph, should cover the evidence that these factors were present in Europe but not in other cultures of the period. You should finish your transition paragraph by pointing forward to the order in which you will address your subtopics.
The first main body section of your paper should cover how Europe fulfilled all three criteria you list. First you should mention that Europe's twelfth century Renaissance began with contacts with Islamic culture in Spain, and that both Islamic science and ancient Greek texts were transmitted to the west in the great translation centers of Toledo just as Islam itself was becoming more legalistic and suspicious of science. Other cultural contacts were the result of the fall of Constantinople, the Crusades, and the Age of Exploration. Next, you should cover the origins and flowering of the medieval universities as faculty run institutions rather than ones with strong separate administrative hierarchies. Next, you might cover patronage of the arts and sciences by the nobility and the church. You should also mention the rise of Protestantism in northern countries, as the Roman Catholic Church did discourage Galileo and other scientists.
The next main body section of your essay should discuss why other cultures such as the Chinese, Islamic, and indigenous American ones did not make similar scientific discoveries. Your conclusion should emphasize that it was a combination of the presence of positive and relative absence of negative factors that led to Europe's development of science.
You might find the work of Thomas Kuhn interesting as a source of insights for your paper.