Paul is here defending his teaching about mankind's sinfulness and guilt before God. He has adopted the role of a prosecuting attorney to prove that everyone has broken God's law and is in need of forgiveness. Knowing human nature, he has anticipated objections and now is going on to prove that the particular objection in view is invalid.
In effect, his point here is that judgement of the world by God is a basic axiom, a first principle of theology. Therefore any idea or argument that would deny Him the right to judge men can simply be rejected out of hand. We may not dismantle God by taking away the fundamental attributes that make God to be God. God will judge the world. Period
But it is common to this day for people to live by philosophies and standards that rest upon the very notion that they will never face God's judgement. But God will not cease to be God or to exercise His divine prerogatives. It is the sinner that must change to escape a disaster that is beyond words to express, falling under the wrath of Almighty God at Judgement Day.
God is scrutinizing the lives and hearts of you and me right now. "The LORD is in his holy temple, the LORD'S throne is in heaven: his eyes behold, his eyelids try, the children of men" (Psalm 11:4). That's watching and measuring us pretty carefully. The consequence is sober indeed. "Upon the wicked he shall rain snares, fire and brimstone, and an horrible tempest" (Psalm 11:6a).
But none of this is revealed to us to bring us to despair. It is far better to know our danger now, so we can do something about it. The next thing for us after death is the judgement. It is better to deal with the problem now while we may than to think of it with terrible regrets for ever in hell. No matter how busy we are or how frightening the topic seems to us.
When Jesus Christ, God's Son died on the cross, He already suffered the penalty for our sins. Now risen from the dead, He invites us to turn from our sins and receive Him personally as Lord and Saviour.