Quick answer
The book's central posture produces confidence rather than anxiety, which is the opposite of what "fear" suggests in English.
Read both lines
This is where the book's own definition matters most. If the fear of the LORD were dread, the verse would be incoherent, because dread does not generate strong confidence. Read with 8:13, where fearing the LORD is hating evil, the sense is straightforward: a person whose loves and aversions are rightly set is not braced for anything. The second line extends the shelter to children, which is how this book usually thinks about time, and the word for refuge is a place you go rather than a feeling you have.
Related: Proverbs 8:13 · Proverbs 18:10 · Proverbs 14:27
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.