Wrapping up my comment as an answer:
Imagine you’re a Japanese cryptanalyst in the year 1944. There is no such thing yet called “television”, and you’re still decades away from a wordwide network feeding you with all the knowledge you could wish for.
In that case there’s only a minimal chance you’ve ever heard or seen a Navajo. So, you’ll be wondering what “Joe” is talking about as it sure doesn’t sound like English. You‘ll be asking yourself things like: Is that the plaintext in some language I don’t recognize or simply a failed decryption attempt?
Furthermore, even if you would know it is Navajo and assuming you would have some plaintexts to your availability, you would still be wondering about the more simple things of cryptanalysis, like: What’s the character frequency of Navajo anyway?
Even more, since Navajo is a descriptive language. Even having things like character frequencies and plaintexts at hand, how would you know what is meant by (eg) “wolf feeds bisons when eagle moon embraces cloud rock”?
See, Navajo indians may know “cloud rock” from stories told by their elders and they may know the exact day in the year when the “eagle moon embraces“ it… but you – sitting in your cryptanalysis room in Japan – have a rather minimal chance of even grasping half of it.
All in all, the use of Navajo boils down to a suplemental layer of “security through obscurity”. Maybe it helps if you think of the Navajo communicators as “living codebooks”. Embedding them (especially in key positions) added an additional layer of security to the already used cryptographic protection during radio communication et al.
In fact, take the actively used (radio) encryption during WW2 and add a layer of Navajo language use on top of it, and you’ll end up with something that could almost be called “double encryption”.
To answer the additional question from your comment (now follow-up question added as an edit)…
I think both of these are good explanations of why this scheme worked. From your explanations it seems to me the primary reason it wasn't broken is because it happened 70 years ago. Am I correct in saying this would never work today?
Under most conditions – Yes, that’s correct.
Of course, this depends on the exact situation. The reason I’m saying that is that there aren’t many Navajos living today (last I heard was something in the thousands). Some specific scenarios might change that “yes” to a “no”. But generally, looking at the means (and resources) available today, I tend to say that – in contrast to the WW2 – they indeed wouldn’t be very helpfull in terms of security. Actually, it’s like using the Klingon language from Star Trek, or Elvish from Lord of the Rings… as long as only a few know it, it might add a minimal bit of security. But nowadays, you can even take online courses to learn Klingon, Elvish, etc. Back in 194-something, the situation was quite different.
As an aside: I used the name “Joe” on purpose, since it was a commonly used nickname for US-American soldiers during World War 2. Another example of such a naming would be “Tommy“, which was used to hint at English soldiers.