First, harking back to 1:6—2:2, John writes because your sins have been forgiven on account of his name. On the one hand, this statement serves as a reminder of the way in which they entered the community, namely, through confessing their sins and receiving God's forgiveness in Christ. On the other hand, the statement also speaks of the present reality of the need for forgiveness that is readily and continually available.
The phrase on account of his name refers to the "name" of Christ. In Semitic thought the "name" stands for the whole person. To "believe in the name of the Son of God" (1 Jn 5:13) means, quite simply, to believe in Jesus himself. Similarly, forgiveness on account of his name means forgiveness for his sake, or for the sake of what he has done. What is in view is primarily the atoning sacrifice of Jesus' death on the cross (2:2). Because of Jesus' sacrificial self-giving, God forgives sins through him. Those who have experienced this forgiveness are the children of God, the author's own dear children, the congregation of faithful believers.