According to an interview with Star Trek Magazine, the show's co-writer Joe Menosky came up with the idea of a metaphor-based language from two key concepts; the linguistic notions of John Ciardi (that every single word has a storied etymology that must be understood before a translator can accurately translate a work into another language) and the "imagistic shorthand" used in Ancient Chinese speech and writings (where short phrases are used convey meaning that is unfathomable to people who hadn't been brought up to understand what the speaker is alluding to1)
Change of direction
“Fortunately, Michael went off to see ‘Dances With Wolves’ and came back to the office completely taken by the scene of Kevin Costner and the Native American by the fire, where they try to communicate with each other. He told me he wanted, ‘Two people, on a planet; they don’t speak the same language, but after a great struggle they finally break through to understanding.’ And I said, ‘I can do that.’ So I threw out the original script, kept the title and came up with the story Michael wanted. If he had not seen 'Dances With Wolves’ that weekend, ‘Darmok’ would never have reached the screen, and I may well have been out of a job.”
One of Joe’s first practical problems was developing a form of language that Picard wouldn’t be able to understand. In STAR TREK the crew happily travel around the Galaxy encountering countless races who appear to speak English; this is explained away by the use of the universal translator an almost magical device that can instantaneously translate any language. Somehow, Joe’s aliens would need to speak a language that baffled the technology.
“Our understanding of the universal translator at the time was pretty vague,” he says. “No episode had unambiguously established what was going on, and nobody on the staff had it worked out. I assumed it used a vast database, with hundreds of thousands of languages and some sophisticated knowledge of deep grammatical structures common to all humanoid life forms.
“The problem I had in terms of story was that I wanted Picard and Dathon to actually speak to each other, rather than try to communicate through gestures or miming. But what the alien was saying had to be meaningless to Picard, or else the story Michael wanted me to tell could not happen. I needed an informing concept. The poet and translator of Dante, John Ciardi, once wrote ‘every word is a poem’ 2 - meaning that if you look into the history of any word you will always get back to a metaphorical image.
“I combined that notion with the kind of imagistic shorthand sometimes used in ancient China: like ‘Viscount Yi.’ If you don’t know that Yi was a minister at the court of a madman and what he did to survive, then you don’t know what that phrase is supposed to convey. So that was the scheme I came up with; the Tamarians speak exclusively in metaphoric shorthand, based on their own history and their own myths. And if you don’t know those stories then you don’t know what they are trying to say, even if you understand each word in isolation, which is all the UT could give you.”
Star Trek Magazine: December 2002 Volume 3 Issue 8
1 - In much the same way that from the Western tradition might say "it's Pandora's box" to allude to a situation with unknown but almost certainly bad consequences, referencing Hesiod's Works and Days.
2 - Technically speaking, Ciardi said "every word is at root an image, and poetic images must be made of words" not "every word is a poem" but the meaning is basically the same.