Is the "really" in "I don't really know" necessary?
Solution 1:
Frequently, it is simply a matter of preference. However, the following variations could be interpreted slightly differently.
I don't know.
A simple declaration. Unambiguous.
I don't really know.
This is often used to deflect a negative self-assessment. It's common to not want to admit that we don't know something, so we occasionally de-emphasize the declaration.
I really don't know.
You didn't mention this variation, but this is often used to do the opposite, actually emphasizing the fact that we don't know. This can be used to stress to someone who may not believe you, that you in fact, really don't know something.
These are really just minor nuances, and the phrases are often used interchangeably. If someone wants to express a degree of knowledge, some other phrase is usually used in conjunction with the declaration. For example, "I don't know enough about X to answer that."
Solution 2:
I don't really know suggests that the speaker may know a little about the subject discussed but not really that much.
For example,
I don't know Jimi
probably means you have never heard of Jimi, while
I don't really know Jimi
means you may have heard of or even met Jimi but you do not know much about him.
Solution 3:
Different languages emphasize different things. Japanese, for example, has a tendency to emphasize the speaker's belief in the probability of events and the certainty of knowledge. The number of ways a Japanese speaker can say "I think it will rain tomorrow" is quite large.
English, in contrast, is very sensitive to order in which things occur or are likely to occur, as evidenced by such things as the future perfect tense (e.g., "by then I will have known the result).
But English also has a strong bias toward stress, so that degrees of knowledge are able to be pinpointed: I don't know. I'm not sure. I sort of know. I guess. Maybe. It's possible that... Etc.