Paul’s talking about the relationship between a believer in Jesus, and an unbeliever, the relationship between a found person and a still lost person. It may seem strange to you, but by virtue of you being saved by Jesus, and so being a bearer of the gospel message, the follower of Jesus has inherent power and authority.
It’s the power of the gospel. We are not ashamed of the gospel, because it is the power of God for the salvation of all who will believe. It’s a power the follower of Jesus must exercise carefully and responsibly, because so much is at stake.