Quick answer
Money quiets people, and the verse reports that it works. It does not say it should.
Read both lines
Both halves describe the same transaction at two temperatures: a gift for anger, a bribe for strong wrath, with the payment escalating alongside the fury. The NKJV footnotes "behind the back" as literally in the bosom, the fold of the garment where a concealed payment travelled, so the concealment is written into the Hebrew rather than added. What makes the verse uncomfortable is that it is not framed as a warning. It is an observation, delivered flat, and the book supplies the verdict elsewhere: 17:23 says a wicked man takes a bribe from the bosom to pervert the ways of justice. Read together they say the mechanism is real and the use of it is corrupt.
Related: Proverbs 17:23 · Proverbs 17:8 · Proverbs 18:16 · Proverbs 15:27
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.