These verses mean:
"Having purified your souls by your obedience to the truth": Peter is reminding his readers that their souls have been purified by their obedience to the truth, which is the gospel of Jesus Christ. This obedience involves repentance from sin and faith in Jesus as Lord and Savior.
"for a sincere brotherly love, love one another earnestly from a pure heart": The obedience to the truth that Peter speaks of is not just about personal salvation, but it also leads to loving one another earnestly from a pure heart. This is what characterizes sincere brotherly love.
"since you have been born again, not of perishable seed but of imperishable": Peter reminds his readers that they have been born again by the power of the Holy Spirit, and this new birth is not from perishable seed (i.e., human beings), but from imperishable seed (i.e., the word of God). This emphasizes the eternal nature of salvation through Jesus Christ.
"through the living and abiding word of God": Peter emphasizes that this new birth comes through the living and abiding word of God, which is the gospel of Jesus Christ. The word of God is living and active (Hebrews 4:12), and it has the power to transform lives and bring about salvation.
Overall, 1 Peter 1:22-23 teaches that believers should love one another earnestly from a pure heart, since they have been born again through the living and abiding word of God. This new birth is the result of obedience to the truth, which involves repentance and faith in Jesus Christ as Lord and Savior. This new birth is what enables believers to live in love and obedience to God and to love others in a sincere and genuine way.
1 Peter 1:22-23. Seeing you have purified your souls in your obedience to the truth through the Spirit in sincere brotherly affection, love one another from the heart fervently: having been born again, not of corruptible seed, but of incorruptible, through the word of God, which lives and remains forever.