In this chapter of his letter, Paul is defending his apostolic authority. There were those who objected to his teaching and the Gospel that he preached, and so to undercut him they attacked his position as an Apostle. Paul responds that he is not affraid of their attacks, because he knows that only God can judge him. Besides which, the mark of his apostolic office is that God is working through him and the Gospel with power, not just fancy talk. This is a subtle counter-attack, because he's stating that his opponents do not demonstrate the power of God and rely only on rhetoric and personal attack to establish their authority.
2 Kings 2:1-14) - talk about making an exit! The other event in his life was the showdown between Elijah and the prophets of Baal (1 Kings 18:1-40). Ahab, the evil King of Israel, and his wife Jezebel, had led Israel astray and caused them to worship Baal instead of God. So God sends Elijah to get everyone back on the right track, and Elijah proposes a contest to see who is the real God. Two altars will be set up, and the prophets of Baal will call out to their god, and Elijah will call out to his God, and the true god will be the one who demonstrates their power by setting the altar on fire. Simple. Very definitive in the results. In fact, why any religious debate since then has been settled in any other way is beyond me.
God demonstrates his power by setting the altar on fire. Everyone saw the power of God manifest in space and time, and they were faced with a decision. They could either try to ignore it or explain it away, or they could trust God and repent and return to Him. We Christians believe that God has in fact acted in a supernatural way in space and time. God's greatest demonstration of his power is in raising his Son Jesus from the dead. You have to deal with that power - either you ignore it or you trust it. Jesus even says to his detractors, hey, you might not like me or what I'm saying, but if nothing else believe me because you've seen the miracles (i.e. the power I've demonstrated).
Where in your life do you need to experience the power of God? Where do you need to know God's forgiveness? To be healed? And we do hope that the power of God can and will do something, for God whose power is at work in us is able to do more than we can ask or imagine.