After his greeting, Paul starts his letter to Philemon praising him for how he shows his faith and love¹ in the Lord by serving his fellow believers. He uses this as his foundation for his coming appeal for Philemon to continue his acts of service and kindness by freeing Onesimus from slavery and accepting him as a brother in Christ.
The basis of Philemon’s actions toward others stems from his faith in Christ and his love for people. This has a direct correlation to what Jesus said were the two greatest commandments: “Love the Lord your God with all your heart, with all your soul, and with all your mind [and] love your neighbor as yourself.” (Matthew 22:37,39 HCSB) Likewise, it is the basis of his appeal for finding favor with Philemon.
Paul prays that Philemon’s partnership with him in the faith helps Philemon understand what Paul is going to request from him.² In the same way, we all share in the faith and our prayer should be that it helps us come to a better understanding of who God is and share in “every good thing” (verse 6). This can only come through sharing our faith and our bond of love for each other. We can’t stand alone. It affects how we live and the actions we have with our fellow Christians. The more we share in the faith, the more we love each other, our walk with God will deepen–move to the center–and we will be what God calls us to be.
“A new command I give you: Love one another. As I have loved you, so you must love one another. By this everyone will know that you are my disciples, if you love one another.” John 13:34-35 NIV
¹ In the Greek, the words Paul uses for love and faith in verse 6 are agape and pistis, respectively. Agape here is a reference to human love, that is, “the quality of warm regard for an interest in another; intimate relationships without limitations, [but not sexual].” Pistis here is a reference to faith in Christ, that is, “a state of believing on the basis of the reliability of the one trusted; faith in the active sense…in ref. to deity;” In this context it is specifically denoting a “belief and trust in the Lord’s help in physical and spiritual distress.” (Taken from A Greek-English Lexicon of the New Testament and Other Early Christian Literature (BDAG))
² Arthur G Patzia puts it this way: “The best way to approximate what this verse  means is to see it in context of the letter as teaching these truths: All Christians share a common faith; faith should be and active faith because it promotes an understanding of the blessings believers have received; the response of faith is for Christ, that is, for his glory; Philemon’s recognition of these blessings will cause him to respond appropriately with respect to Onesimus.” (New International Biblical Commentary: Ephesians, Colossians, Philemon, pg. 109)