Paul does not stop at advising Philemon from afar, he promises to visit. This, more than anything else, indicates a strong friendship between these two men. Given that Philemon is a leader in a local church Paul probably planted, it is likely they had endured persecution together in the service of Christ. So Paul asks to come stay with Philemon, and probably for no small amount of time given that he is asking for a room.
Paul knows Philemon and everyone else at the church have been praying that he will be released form the Roman prison and free to travel. That is why Paul writes of the upcoming visit as an answer to prayer for both himself and Philemon. Even in his imprisonment Paul’s disciples Demas, Luck, Mark, and Aristarchus are with him. So this suggests it is the first imprisonment, from which he is indeed released rather than the second which ends in his execution.
Paul is asking Philemon to free Onesimus from the burden of slavery and from any debt he owed Philemon. What debt do we hold from which we need to set someone free? Who are we holding captive through manipulation or power? Wouldn’t Jesus want them set free as well?