This is the sentence:
Watashi wa kangokujin de wa arimasen.
I'm not Korean.
I was expecting something like:
Watashi wa kangokujin arimasen.
Why is the では necessary? And what does it mean?
ではありません should be treated as a single word (for now, at least). It is the negative form of です, so it means 'is not', 'am not', 'are not' etc.
You may have already seen the contracted form of the plain speech version じゃない. In this form じゃ is a contraction of では, and ない is the plain speech form of the polite ありません.
On its own ありません is the negative form of あります which is the polite form of ある. This means 'to have', 'to exist' etc. So your sentence 私は韓国人ありません is ungrammatical and doesn't make much sense. 私は韓国人がありません would mean "I don't have a Korean person" which sounds pretty odd to me.