If the sentence was 'I am A man' then the correct answer would be 'Jeg er EN mand' with only 1 n. However the sentence is 'I am THE man' then the correct translation is 'Jeg er mandEN'
It is not a very common sentence but a grammatically correct sentence anyway. But in this case I would guess it to be because you need to learn how to qualify the word and make it a specific person (the man = manden) instead of a general person (a man = en mand)
What an interesting thing I learned reading the comments on everybody. Danish is more easy than I though... its kinda bit weird when you say: oh, there are some words that dont exist in Danish, but then you realize that is better because that way you learn more faster. Great people at below's comments :)