When Ephesians 3:10 tells us that "through the church the manifold wisdom of God is made known to the rulers and authorities in the heavenly places," it is telling us that the church has a unique place in God's economy to show something about Himself to not only the world, but also to the angelic host in heaven.
The word manifold in the original language means to be multi-colored. It was used when describing the many colors of a painting or a tapestry in the ancient world. When speaking about God's wisdom it describing the many angles, hues, and dimensions of God's knowledge and its application to all things.
God's wisdom is one of many of His attributes or characteristics. God is loving, merciful, all knowing, all present and all powerful. God is just, faithful and perfect. He is eternal and infinite. God is also full of knowledge and wisdom. Wisdom is God's ability to apply His knowledge in perfect measure which always has goodness and His glory in mind.
Psalm 147:5 describes God as "great, abundant in power, His understanding is beyond measure." or as the KJV describes it "infinite." AW Tozer in his classic book on God's attributes, The Knowledge of the Holy, says that "God does not struggle to apply knowledge in any given situation... God's wisdom sees everything in focus in relationship to all and God is able to work everything out to flawless precision."
How is this wisdom revealed through us, the church? First, the church is a gathering of sinful men and women who have been forgiven and redeemed through the blood of Jesus Christ. It shows God's love and His ability to bring about salvation to broken humanity. When we look at a sinner saved by grace, transformed by the Spirit, we are seeing God's wisdom.
But the church also shows God's wisdom when it becomes an ambassador for Him to bring the message of reconciliation and salvation to a lost world. Sinful man reaching sinful man. "If anyone is in Christ he is a new creation, behold the old has passed and the new has come. And this is through the Lord who reconciled us to Himself and gave us the ministry of reconciliation." (II Cor 5:17-18)
This redemption of sinful man is something that shows God's wisdom not only in this age but also in the ages to come, eternity. It is everlasting, where reconciled men and women will be with Him forever and ever, living out His purposes.
All of this is being revealed through us, the church to "the authorities and rulers in the heavenly places." This is the angels God created. This includes Seraphim, Churabim, Archangels, and Angels. This also includes the demonic realm, including Satan and his demons. They all are looking at God' church, God's salvation in the midst of sinful man and can see "His manifold wisdom."
The application for us is to remember our God's given purpose, which is very great indeed. As we live out our lives in holiness, in a way that reflects God's goodness and work in us, God's wisdom is seen. As we proclaim the message of the gospel as ambassadors for Christ, we are revealing His wisdom not only in the world, but also to the angelic realm. And that is one great purpose to live for!