This packs a whole question inside a sentence and says the speaker has no answer to it: 뭔지 모르다 is "don't know what it is", 어딘지 모르다 "don't know where it is", 몇 번 칸인지 모르다 "don't know which car it is". The question word does the work, and the -ㄴ지 part is what lets a question sit inside a bigger sentence, so you cannot drop it and just say 뭐 몰라요. Speakers very often say 모르겠어요 rather than 몰라요, which softens it to "I'm not sure" instead of a flat "I don't know". Watch for one twist: after 얼마나, as in 얼마나 다행인지 몰라요, it is not ignorance at all but emphasis, closer to "you have no idea how glad I am".
Register: neutral; works in casual and polite speech alike