This is one of those verses that, if you grew up in the Church like I did, you hear often. Typically this verse is used to make the point that God, as God, is beyond us. He is unfathomable. He is all knowing and all powerful. As far as I have heard, and probably said myself, this verse is used to make a general statement about the nature of God. Now, what is being said about the nature of God may very well be true, and I would argue that for the most part, it is, but Isaiah is doing anything but making a general statement about God. If we are going to understand what Isaiah is doing here, we must keep this passage connected with its surrounding context. Here is Isaiah 55:6-7.
Seek the LORD while he may be found, call upon him while he is near; let the wicked forsake their way, and the unrighteous their thoughts; let them return to the LORD, that he may have mercy on them, and to our God, for he will abundantly pardon.
I added the italics to help focus our eyes on what Isaiah is focusing on. Again, Isaiah is not making a general statement about God when he says, "my thoughts are not your thoughts." He is making a specific statement about a specific aspect of God, namely God's forgiveness. Isaiah is saying that God abundantly pardons. He pardons more than we would pardon. He forgives when we would not forgive. And why is that? Because, God's ways are higher than our ways.
To get on a familiar soapbox, when many read the prophetic books in the Old Testament, all they see is a God of judgment. To be fair, there is judgment in the prophetic books, but that does not mean the over-arching narrative of the prophets is one of judgment. That narrative is one of forgiveness. Discipline was necessary for Israel, as it usually is with spoiled children, but the last word is forgiveness and restoration. It is abundant pardon. For it was while we were still sinners that Christ died for us. Paul's thoughts shouldn't be a shocker. That is how God has always been. As the heavens are higher than the earth, so are his ways higher than our ways, abundantly higher.
Stopping point: Isaiah 66