Translation:Why are you not eating breakfast?
Examples of how to use this Japanese:
Child: (sitting at the table with their breakfast untouched)
Parent: Why aren't you eating breakfast?
Why aren't you eating breakfast (every day)? = Why don't you eat breakfast (every day)?
I think duolingo was trying to differentiate from the invitational "why don't you ~?", but it's still not a clear translation because it gets confused with the present progressive. I definitely didn't type the duolingo answer and whatever I wrote was accepted, probably "why don't you eat breakfast?"
I guess I didn't explain very clearly, but the two situations I listed are examples of how the Japanese is used. In situation one where the child is sitting at the table and not eating, it's not unnatural to say どうしてあさごはんを食べないんですか. It does not require the -te form. I often find if I'm at a party and do not eat continuously from start to finish a Japanese person will unfailingly ask me 食べないの？ (tabenai no, you're not eating?) without using the -te form.
Japanese is a very contextually dependent language. The "-n" nominalizes the verb it follows. So, literally, the sentence says, "Why is it that (you) do not eat breakfast?" The "desu" is stative and there is no need to use "-te iru." (Although it probably wouldn't be ungrammatical to do so.) The literal meaning can apply to either context.
The "n" is a form of the particle "no." The particle syntactically turns the verb "tabenai" into a noun. This verb is a non-past form which, as a verb, generally indicates customary state or futurity. Nouns do not have tense. So, "tabenai + n" means something like "a not eating." "Desu" affirms the noun with reference to the topic-subject, which in this case is the unexpressed second person, "you." The result of that, in this case is literally something like "a not eating is affirmed about you." The "ka" makes this a question, "is a not eating affirmed about you" or, a bit more free, "does a not-eating apply to you?" If the "dou shite" connects with "tabenai" you get something like "does a why not eating apply to you? " If it can be argued that the "dou shite" connects with "desu" you get "why does a not-eating apply to you? " Or, more rigorously, "Why is it a matter of your not eating?"
I've heard this explanation a lot but when discussing it with my Japanese professors no one particularly felt that this is the case. Nor did I find anything in grammar books we have here at the university, like the multi-part 現代日本語文法, to support this theory. ～のですか is simply a question asking for explanation, and ～のです is an answer to that type of question.
The way I see it, it might or might not be related to nominalization. But just because it looks the same doesn't mean it is the same. After all, の-nominalization and の-possessive are hardly the same thing. If you read it somewhere, I would very much like to see the source.
The ～のです form changes the meaning of the question a little bit. It's asking for explanation, so instead of a simple yes/no answer, the listener is expected to provide a bit more detailed piece of information. That's why you can never use ～のですか in pure yes/no questions. Answer to this type of question will also contain ～のです, so for example:
～のです has the same level of grammatical politeness as standard ～ます form. However, the very act of asking for an explanation means that you should be at least somewhat familiar with the person and/or be in proper context, otherwise it might be seen as prying - just use common sense. As was said by others, the contracted variant ～んです is a more colloquial, less formal variant.
DL is currently translating "どうしてあさごはんを食べないんですか？" as "Why don't you eat breakfast?" and in English that translation has (at least) two connotations: the first is an invitation like "won't you have some breakfast?" and the second is a straight-forward question similar to "for what reason do you not eat breakfast?" Which of those two is implied here in the original Japanese?
I believe the use of の or ん in this manner (nominalization) is a somewhat feminine usage especially in a question but I'm not certain about the parameters of gender specific usage
A 日本の友達 told me her male friend learned 日本語 from his wife so he always sounded a little effeminate My 日本語 was weak but better than my friend's husband's 英語 so he & I used 日本語 a lot But they were always very careful to correct me if I used a word I heard him use that wasn't considered proper for a female For example I could use 出来る meaning to be able to but not as he used it to do something well or make something That was considered too masculine --I'm really stretching my memory here
～のですか / ～んですか is simply asking for explanation or reasons, it is not feminine / masculine speech by itself, it's a part of grammar.
What might be perceived as feminine or childish is excessive use of ～のです / ～んです, but I'm not entirely sure. I guess it also depends on where your friend is from.