I think it helps prevent ambiguity.
da solo definitely means "alone", while solo can be either an adverb meaning "only/just" or a predicate adjective meaning "alone":
voglio solo stare - "I just/only want to stay/to be"
voglio stare solo - "I want to be/stay alone" or "I only want to stay" or "I want only to stay". Position of solo in the sentence will make a difference, too.
It's also probably an idiomatic format that means "alone" - it's just what you say when you mean "alone" in this kind of sentence.
And I thought my answer 'I want to stay single' was a good guess. I laughed out loud when I was corrected with 'I want to stay lonely.' LOL, LOL, LOL No-one ever says that! The translation here 'I want to be alone' is more like it. You have to admit, that Duolingo is often quirky and fun and we love it. (January 2018)
The infinitive is 'stare' not 'essere'... so logically the translation should read 'I want to stay alone' not 'I want to be alone'. However as 'Sono al verde' translates strangely as 'I am broke', 'stare' could (like 'verde') be idiomatic for 'to be' in this case. Every language has its idiosyncracies... thankfully!