We are not told about Philemon’s response to Paul, however, we can arrive at a reasoning that Paul’s goal was to have Philemon forgive Onesimus for running away. In order to fully benefit from each other as Brother’s Keepers, learn to forgive and encourage the people around you to purposefully live a lifestyle of forgiveness.
We forgive because we have been forgiven. When the Father forgives us of our sins, we are empowered to do likewise- knowing that we are emulating Him in the process.
Matthew 6:12 (NIV)
Forgive us our sins, as we forgive those who sin against us.
It is easier to forgive when you have Christ in you because you understand the significance of forgiveness. You understand the price He paid for your reconciliation with the Father. You will be a better person if you live a lifestyle of forgiveness versus resentment, bitterness, grudges, etc.