"My big brother is tall."
You don't really need the わたしの here - あに implies you're talking about t your own; otherwise it would be おにいさん.
Describing aspects of someones body has a set format in japanese it goes: "aspect が adjective です"
To call someone tall you would say:
せ(Height) が高いです: literally height is tall
Along the same lines: かみがながいです: Hair is long.
It's correct, too. Because of the の nominifier,「兄の背が高い」 literally translates to "My brother who is tall" or "My tall brother". It makes that adjective something your brother owns.
But Duolingo wants to express a subject and a complete sentence with the use of は. And, in that sentence, you see an 'ownership' of that brother of yours (whom you say is tall) because of the の nominifier.
Adding an んです implies 4 things: 確認している; 情報を聞いて; 説明して; and 頼んでいて.
Literal translation of that sentence is "Because my brother is tall". Could this person be bothered by the fact that their brother's tall?
わたしの should be optional, it is probably even more natural to just leave it out.