The second part of Proverbs 3:5 says:
"and lean not unto thine own understanding."
The New King James version calls it "insight".
I suppose the way to follow this second part of the passage (Proverbs 3:5-6) is to follow the first, first. Another word for "lean" is "trust". And if we're trusting in God (with all of our heart/life), then we'll inevitably run up against a clash where the Christ-like response that we know we should have, either from having read the Bible, or having received an intimation of His character directly, seems to contradict the situation at hand and the "correct" response it calls for.
"Set a watch, O Lord, before my mouth; keep the door of my lips." (Psalm 141:3)
I'll give you an example: say you have a friend who is clueless to something that you see, and in your opinion "everyone sees". First of all, God sees everything. And if you really are seeing one layer deeper about a subject, then it's God who has allowed you to see it in the first place (this is His insight). But put this into practicality. You, in and of yourself, are unable to open their eyes—to remove the scales, as it were—on your own. This would be an example of your own insight clashing with God's. When I think that I can, just by informing my friend of this deficiency in his outlook, open his eyes and make him see what I think he needs to see, I'm relying on my own insight. And if I continue on, I could block the hand of the Lord from revealing to my friend what he needs to see and I could even end up hindering the Lord from healing the breach in his heart that caused the blindness in the first place. The danger in revealing things to people that they themselves are blind to is that they could end up trusting you and not turning to God for themselves. The correct response in any situation and especially those involving people—friends, family—is to trust God and not rely on your own insight. Hold them up in prayer. "Bear the burdens…" (Galatians 6:2)
Love builds up, it "edifies" (1 Corinthians 8:1). Forgiveness and compassion are aspects of love that, when lived out towards God, ourselves and others, circumnavigate the harsh, ineffective and ultimately detrimental insight that (I believe) Solomon is referring to in this verse. The same idea is expressed by Paul in the first letter to the Corinthians (8:2) when he says "and if any man think that he knoweth any thing, he knoweth nothing yet as he ought to know." He continues in the next verse: "but if any man love God, the same (that person who loves God) is known of Him." and again, same book (10:12), "let him that thinketh he standeth, take heed lest he fall".
One of the best ways to see our own insight in action is to observe people, faults and all (knowing we ourselves are the same—again, Galatians 6:1), and consciously make a decision to see past our old, own insight and see them as God does. Forgive them. Love them. Believe the best of them. "Love is ever ready to believe the best of every person." (1 Corinthians 13:7 AMP)
God bless you!