One of the meanings of "salt" is to spice something up or make it more tasteful and lively. If you salt a story, that means you make it more interesting. "Tell me the salt of the story." means tell me the interesting parts.

I am a native english speaker and I have NEVER heard this turn of phrase. I would assume that it means the most interesting part of the story to the person you are speaking to. This is an important distinction as it may not be the part of the story that is most important to you.

It means the only most important parts of the story! Marco