Love (8-9): Paul had the authority to call believers to action,[1] and he could have ordered Philemon to forgive Onesimus. But given Philemon's long track record of love,[2] Paul instead asks him for a favor, not as an imposing apostle, but as someone helpless ("aged" and "a prisoner"). He is sure Philemon will treat him (and Onesimus!) with this love. Christian love is dependable.
Onesimus (10-12): By God's providence, Onesimus met Paul in Rome, and he became a Christian.[3] But previously, he was only a "useless" slave. He cheated his master of rightful service (by running away), and he also wronged his master financially.[4] But upon conversion, he became "useful." In Rome, he faithfully ministered[5] to the imprisoned Apostle and endeared himself to Paul and became the object of his deep affections.[6] Paul was convinced this man would prove useful to Philemon also. God can change someone into a blessing.
Giving (13-14): Paul would not keep Onesimus with him without his master's knowledge and consent. This man was legally still Philemon's possession. Would Paul then demand that he offer his slave's services? No. Such goodness ought to arise from one's own volition and goodness, not from coercion. Paul leaves it up to Philemon to decide. Giving should be voluntary (2 Cor. 9:7).
[1] This authority is seen in 1 Thes. 4:2, 2 Thes. 3:6 and 2 Cor. 13:10. This wasn't only apostolic (cf. Tit. 2:15).
[2] "For love's sake" is literally "on account of the love," which refers to Philemon's love (1:5, 7). This type of Christian love is expected of all true believers (cf. 1 John 3:14; 4:7-8).
[3] Cf. 1:15-16. The language of spiritual fatherhood is common in Scripture (cf. 1 Tim. 1:2; Tit. 1:4).
[4] 1:18-19 indicates that he wronged him and owed him a repayment, which Paul offered to pay in his place.
[5] The grammar of 1:13 points to a service already in progress: "I was wanting to continue keeping him so that … he would continue to minister to me." Believers commonly cared for the saints in prison (Heb. 10:34; 13:3).
[6] The phrase "my very heart" is my very guts (splangch-non [σπλάγχνον]). This speaks of deep affections.