According to most linguists, Korean is a language isolate. Why doesn't it have any sister languages, like languages usually do? Why didn't it spread to other areas, or split into various languages? The speakers are not geographically isolated from other areas (unlike Japan, for example), and they also have relatively large number of speakers. What are the historical factors that caused its isolation?
I think this question represents a certain confusion of terminology. By definition, a living language does not have daughters (excluding some circumstances where a parent language is retained in a particular use after the spoken languages have moved on and split, as with Latin, Sanskrit, Classical Arabic, etc.) So the question should not be "Why doesn't Korean have daughters?" as much as "Why doesn't Korean have sisters (other languages descended from the same parent)?"
With that in mind, here are some points:
- Korean does exhibit dialectical differences. It isn't exempt from the process of language change.
- Modern Korean is descended from Old Korean, which was probably one of the Buyeo languages. As the name suggests, there were several languages spoken in the Korean peninsula at this time, which are believed to be descendants of the unattested Proto-Korean.
- The fact that there is only one language currently spoken on the Korean peninsula appears to be because one dialect gained enough prestige to eliminate its competitors. In other words, Modern Korean is an isolate because it killed off its sister langs. The timing of this sororicide is unknown, but probably occurred during the unification of Korea in the Middle Korean period (my speculation).
Edit: One more thing: I doubt whether it's actually a case for a language to "normally" have sisters or daughters. All languages change, but language change only results in language split if the original speaker community has spread out enough to lose frequent inter-communication. The number of languages that expand rapidly in this sense is actually rather small, though the fact of their expansion means that they represent a large fraction of all language speakers.
So it seems most accurate to say that most languages have no daughters (or only one daughter, depending on how you look at it), but that most languages are descended from languages that have many daughters. In this sense, language isolates are "normal", while the big language families that we're familiar with are the result of some "abnormal" period of historical expansion.
My understanding of a language isolate is a language that has no relationship with another language, which means that it does not share a common ancestor with another language. (JSBãngs said it well above.)
KO may be an isolate, it may be a member of the Altaic family, it may part of a Sprachbund, or a combination of these. (It's not unheard of that sometimes languages that share a common ancestor later are part of the same Sprachbund too, like in the Balkans.)
The World's Major Languages gives a concise historical background of KO language and the theories of its origins here. (It's very interesting to me because in addition to discussing the linguistic features that KO shares with Dravidian languages, the Austronesian languages and the Altaic languages, it also mentions anthropological and archaeological evidence for each of these theories.)
In short, its siblings, if it had any, died out.
I was under the impression that all languages are considered isolates until it's demonstrated to a high degree of certainty that they're related to another language. Was I wrong on this?
There've been various attempts at linking Korean to Japanese and Ainu, on one-hand, and to the Turkic-Mongolic languages on the other. Should the proponents of one of these efforts succeed, then Korean would stop being classified as an "isolate", which really means "not enough data to accurately say if it's related to something else or not".
It is not language isolate. It is related to Japanese and Ainu and the three along with others belong to Altaic family.