Quick answer
Wisdom sits higher than the fool can reach, and the second line shows where that becomes visible: in public, where decisions are made, he has nothing to say.
Read both lines
The gate was not an entrance in the way we mean it. It was the courthouse and the council chamber, the open space where elders heard cases, contracts were witnessed, and a community's business got settled. So the second line is not saying the fool is quiet, since this book's fool is rarely quiet. It is saying that in the one room where speech has consequences, he is not a participant.
The first line does the diagnostic work with a spatial image, wisdom as something too lofty, out of reach overhead. That is a gentler explanation than stupidity and a more damning one, because what puts it out of reach is not capacity. The fool has been offered wisdom repeatedly in this book and has declined it, so the height is the accumulation of those declines. The saying quietly implies its opposite too. Whoever does speak usefully at the gate got there by a road this man refused.
Related: Proverbs 22:22 · Proverbs 31:8–9 · Proverbs 1:20–21 · Proverbs 17:16
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.