How did all the Nordic Languages (Swedish, Norwegian, and Danish) come to be? I know they all came from Old Norse, but why did they split into these languages?
Languages naturally diverge over time. That's why there are dialects. Over each generation, the language changes a tiny bit.
When people speaking the same language live apart from each other, over the generations the language of the individual groups tends to drift apart. Sometimes there are external influences on one group that are different from those on other groups, too. Or, as with Icelandic, there may be little external influence, so the language tends to drift less. I was just reading about this last night. Supposedly the languages were very similar still during the middle ages. For instance, "in the 13th century there still did not exist a separate term for language of the people of Iceland. The common language of the Icelandic and Norwegian people was still called the 'northern language' (norrǿnt mа́l). The expression 'Icelandic language' (íslenzkt mа́l) appeared only in the 15th century" (Древнеисландский язык, by М.И. Стеблин-Каменский, p. 9). I bet that Wikipedia will have some information about this.
Interesting! I think people also referred to languages more broadly back then. For example, in the 15th century, English was still using both "Eggs" and "Eyren"; William Caxton, one of the first printers in English recounts an anecdote where merchants from the north of England are trying to order "eggs" for breakfast in the south, and get the reply "Sorry, I don't speak French". http://www.bl.uk/learning/timeline/item126611.html Ironically enough, it might be that the distinction between the languages was as much to do with standardisation as it was to do with divergence. If every city has its own local dialect you can call the whole thing "Nordic", but if Göteborg sounds like Stockholm and Odense sounds like København, it starts to make more sense to refer to two separate languages.
> If every city has its own local dialect you can call the whole thing "Nordic" . . .
I doubt that the author of the book would grant you that supposition. England, which spoke Old English, had, after all, been conquered by people speaking a Romance language, Norman French. Such a conquest did not take place in Scandinavia.
When discussing the settling of Iceland (which began in the 9th century) he admits that at that time there were some dialectical differences across the Scandinavian region, " . . . within the boundaries of Scandinavia there already then existed dialectical differences; in particular language differences had already taken shape between the western Scandinavian tribes that made up the Norwegian people, and the eastern Scandinavian tribes that made up the Swedish and Danish peoples." But he goes on to say, "It must be pointed out, however, that the language differences within the confines of Scandinavia were then, it seems, still so small that they were not recognized as differences between languages [не осознавались как различия между языками]. Even significantly later the languages of all the Scandinavian peoples together were called the "Danish tongue" (Old Icelandic dǫnsk tunga). The author of the Icelandic 'First Grammatical Tract' [middle of the 12th century], for instance, calls his language the 'Danish tongue.'"
Between the language of Norway and Iceland the author speaks of differences increasing during the middle ages. He takes it as a given that Iceland and Scandinavia originally spoke the same language: "Over the course of time, the inhabitants of Iceland became isolated from the inhabitants of western Norway, with whom they originally were identical. By the 13th century, when the inhabitants of Iceland had already formed a national character separate from the people of Norway, between their tongue and the tongue of the people of Norway had arisen a few, however very insignificant, differences (basically phonetic [and he give a reference to a section of his grammar]). To the insignificance of these differences in particular testifies the fact that in the 13th century there still did not exist a separate term for the language of the people of Iceland."
Of course, this book is more than 50 years old, so opinions may have changed, and only written records exist as evidence, anyway, so who knows how much the spoken language differed across the area? Still, this is very interesting, and I wish the introductory chapter had been longer.
Some linguists also claim that Skandinaviska is basically one language with four standard forms: Swedish, Danish, Norwegian Bokmål and Norwegian Nynorsk (Icelandic apparently has become too distant to be understood by others).
As others said, when people live in separated areas it is common for languages to diverge, first to dialects and over generations to languages. The difference between languages and dialects can be quite hazy and in many cases strongly disputed (a general rule is that of mutual intelligibility but that can be vague...) and there are several languages with dialects further apart than the Scandinavian languages...
Also, it has been noted that some languages seem to morph faster than others and I don't really know if any good explanation has been found on that (e.g. Icelandic, Finnish and Lithuanian have been mentioned as very conservative, slowly morphing languages compared to their related languages...)
Another question is standardisation of language, which is partially tied to politics: while different places within a country or group of countries have their own dialects, at some point official communication and large part of media starts using a specific form of language with its specific rules, and it is common that people within the country will know both the standard form and the local dialect and can use both as needed (everyone can speak their native language, but it is the duty of education system to teach them the standard form).
In many languages a specific dialect is used as a base for standardisation, often the dialect spoken in the political centre or capital, or in some cases cultural centres. In some languages some sort of amalgam of different dialects is formed, the standard form uses parts of several dialects.
This is a big part of what has happened in Scandinavia: Sweden, Denmark and Norway have each created their own standard versions of closely related languages/dialects, Norway has actually done it twice.
In a similar fashion British English and American English also have a number of different standards even if the mutual intelligibility still remains much higher than the Scandinavian languages. And some other versions of English might also make their demands for their own standards, partly depending on linguistics and partly on politics. It wouldn't be impossible for e.g. Scottish (and not talking about Gaelic) to make a break (and in history of linguistics there has been periods when it has been considered a separate language), and there has also been some political controversy about Ebonics...