How should I interpret a promising preprint that was never published in a peer-reviewed journal?

There might be any number of reasons. You might try to contact the author(s) to get more information. But... (not all with the same likelihood)

They might have left academia for various reasons and not bothered. Is the CV also old?

They might have incorporated the key ideas into another paper with a very different title. You search is then fruitless.

They might have discovered errors.

Reviewers might have considered the results trivial.

Their attempts to publish might have been rejected by journals for other reasons.

They might have changed sub-fields. (This one less likely, I think.)

But you should be wary, at least, of following up on unpublished work and, at least, be sure that you can verify the claims independently.

Not all peer-reviewed papers are solid, and not all non peer-reviewed papers are unsolid.

Judge for yourself.

Seriously, sometimes people cannot be bothered to fight with reviewers about minutia, relevance, impact, significance; worse, sometimes people have a problem to get a paper published in a journal that later proves to be seminal to a field. The story of Schechtman comes to mind (or also some colleague from my own field who wrote an absolutely central paper for my field which took several years to get published in a peer-reviewed journal).

If it is an experimental paper and hard for you to verify, you may tread more carefully, but anything that's theoretical and in your reach to check for yourself is worth consideration if you need it.