No honestly, 'The man needs other people' is really not correct. Unfortunately it is just one of the peculiarities of different languages that sometimes something simply cannot be translated word for word.
The problem is that "Der Mensch" in German is more of an abstract concept than necessarily one specific human being (which it also can be, but just not here). In this specific case singular "Der Mensch" is best translated as plural 'People' and I have to say Duolingo's given answer here is just not helpful or the best.
A fine idiomatic translation in English would be 'People need other people', which conversely could be translated fine as 'Menschen brauchen andere Menschen'.
I know, it's tricky but that is just the way it is. And believe me in this case it's got nothing to do with social engineering. 'Der Mensch' has been 'People' or article-less 'Man' long before the PC-brigade started interfering in logic.
Man means human as in mankind and humankind are synonymous. The man is not the same as the human, the human could be female, the man could not be. I believe the given answer is wrong, it should translate to man needs other humans. 'The man' can only refer to a specific man, which i don't believe this sentence would translate as
Perhaps it is not obvious to someone who is not familiar with either Shirley Bassey or Barbara Streisand: There was a song "People" written for a play (1964?? perhaps) and some of the lines is "People - People who need people - are the luckiest people in the world". This is such a well-known saying for someone who lived through the 1960s and 1970s - the song was very popular, and for most people is a lovely memory. It has little, perhaps nothing to do with whether people would normally say such a phrase. It's art (of the sub-type: MUSIC). Sing along now... "Humans who need humans are the luckiest humans in the world"