I don't get the phrase "a word to the wise". Shouldn't it be "a word from the wise"? Isn't the person with the word the one with the wisdom? Isn't the person receiving the word the one in need of it?
As John Lawler says, the full form of the phrase is a word to the wise is sufficient. A word to the wise is usually employed as a discreet warning in which the ‘word’ is very brief—either the utterance which immediately follows or this utterance itself.
The sense is “I need say no more than a word to alert you—you’re smart enough to understand what I leave unsaid.”
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It implies in a backwards way:
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The context is, "I am warning you about a dangerous condition."
"A word to the wise" means, "A wise person will know what I mean, as soon as I say "beware."
A foolish person will need a lot longer warning/explanation than just one word.
Here's a similar expression