‘Seeing that through the proving (or ‘approval') of you by this ministration they glorify God for the obedience of your confession to the gospel of Christ, and for the liberality of your contribution (‘contribution in fellowship' - koinonia) to them and to all, while they themselves also, with supplication on your behalf, long after you by reason of the exceeding grace of God in you.'
For this ministration of generous giving will be the proof, in the eyes of all, and especially of the Jewish church, of the truth of their own profession of obedience to the Gospel of Christ, resulting in approval of them, and will thus produce the glorifying of God for what He has done in them. And it will also produce thanksgiving to God for the very benefits themselves, and the genuine fellowship that is revealed by them.
Furthermore it will result in the recipients praying for them, and ‘longing after them' (feeling well disposed towards them and wanting more fellowship with them) because of the large amount of the grace of God that it reveals in them. The vision is of the fulfilling of Old Testament prophecies which portrayed ever growing good relationships in God between Israel and the nations (Isaiah 27:13), with both benefiting from each other, fulfilled in Jewish and Gentile Christians being blended together in the new Israel (Ephesians 2:12-22).
Thus they should see that by giving generously they will not only be relieving need, but contributing to the expansion of God's ultimate purposes in many ways.