I'm no historian, but one of the main reasons for many languages, and, in turn, many dialects within a language, is due to the relative distances between places at the time (horse riding speed at best), as well as there not really being a need for travel, apart from the occasional trip to a market. Many people in Europe in the Middle Ages died in the same place they were born, without ever travelling for more than a day from said place.

This limits the interaction between people speaking slightly differently, and, as happens in evolution, eventually causes two different languages to form. Dialects and accents form in a similar manner, but on a smaller scale. Dialects would often be regional variations within a particular language, and towns and villages would each have their own accent.

Actually, Europe is one of the areas of the world with the least linguistic diversity:

Out of Ethnologue’s 6,909 [languages], only 230 are spoken in Europe, while 2,197 are spoken in Asia. One area of particularly high linguistic diversity is Papua-New Guinea, where there are an estimated 832 languages [belonging to 40-50 different families] spoken by a population of around 3.9 million.

It just seems as if Europe has many different languages, as Europe is made up of a lot of very small nations, but many of those nations are only made up of a few languages at most (not including more closely-related languages), whereas most nations in Asia and Africa will have at least tens, if not hundreds, of languages spoken within their borders.