v1: The promises of God's blessings in our lives help us to break from sins. All filthiness must be cut out of our lives. Filthiness of spirit perhaps refers to idolatrous religion. Since we have reverence for God, we seek to perfect holiness in our lives. The clear expectation is that God's people are holy.
v2-3: Following on from 6.11-13. Paul encourages the Corinthians to receive him. Did they not remember his commitment to them? Paul had a deep affection for them, did they not realise that? His love for them was not less just because they had mistreated him. See 12.15.
v4-5: Paul had previously received comfort in Macedonia, esp. in Philippi, Acts 16.12. God sent comfort in their conflicts and fears, such that Paul was joyful in all his tribulations.
v6: Paul had evidently become downcast. But the Lord knows His own, and brings comfort, esp. through others.
v7: Titus came bringing blessing from Corinth, and assured Paul of their prayerful support. How sad that this attitude had not continued. Their longing and sorrow and zeal were strong. No wonder Paul's joy was greater than ever.
v8-11: Godly sorrow produces repentance, contrast Matt 27.3-5. Paul's first letter had brought sorrow, since he had pointed out much that was wrong. The Corinthians had apparently sought to put things right (v11), and Paul encourages them in this attitude.
Again, as in v7, Paul uses strong Greek words here to describe the Corinthians' eagerness, indignation, desire to put matters right. Relationships matter. So there is obligation to put right what is wrong and to strengthen what is weak.
Paul contrasts worldly sorrow, which brings death, for it offers no hope.
v12: Paul had a concern for the church as a whole, and for the glory of God. He did not take sides in an argument; rather he declared the truth of God.
v13-16: The growing relationship between Paul and Titus and the Corinthians. Titus had been blessed at Corinth, and that was an encouragement to Paul. This provides some insight into the working of the 'apostolic band,' which included Paul and Titus. For the individual church, there was the blessing of having such able men to provide support. See also 8.16-24.