Piper does an excellent exposition:
1) Who are these principalities and powers?
the principalities and powers (of Ephesians 3:10) are supernatural hosts in league with Satan and have a cosmic influence on the course of this age, its people, inventions, and institutions. These are the beings to whom the church is to demonstrate the manifold wisdom of God.
2) What is the divine wisdom the church is to make known to them?
So the mystery of Christ is that in His death on the cross He purchased not just eternal life for individuals who trust him; he purchased and formed a new people, a church composed of Jews and gentiles who are both heirs of God’s promises and beneficiaries of God’s grace. That is the first stage of revelation: Paul receives the revelation of this mystery.
3) How are we to make it known?
The wisdom of a plan is seen by the fact that it works. We show the wisdom of God by showing in the church that it is working. The death of Christ was not in vain: it has reconciled us to God, it has broken down the wall of hostility between Jew and gentile and other races, it has produced one new body and it has given us the hope of his immeasurable kindness forever. We show the wisdom of God to the cosmic powers by living this way, by being the church Christ died to create.