Quick answer
Wisdom sits quietly where it lives. Whatever is in a fool gets out, and the contrast is between what stays put and what escapes.
Read both lines
"Rests" is the operative verb: wisdom is at home, settled, not needing to be displayed. The second line does not say the fool broadcasts wisdom he lacks; it says what is in him is made known, which is worse, because he is not performing anything. He is simply leaking. This is the third time the chapter has come at the same idea, after verses 3 and 23, and it is the version that puts the least effort on the fool's part. He does not have to try.
Related: Proverbs 12:23 · Proverbs 13:16 · Proverbs 15:2
New King James Version®, Copyright © 1982, Thomas Nelson. All rights reserved. Full copyright information. Scripture provided by API.Bible.
Go Deeper
If this verse has been used to put weight on you that it was never carrying, ask about it directly.