Because we are the dearly loved children of God, God gives us the gift of his love. A son might expect to receive something from his father, the wealthier the father, the more extravagant the gift. If we have been born again then God is our father and God has access to all of the wealth of the universe, so his gifts to us can be lavish indeed!
Because we are children of God and have received the love of God in this way, the world does not know us (3:1b). The world did not understand Jesus in the Gospel of John, there are many examples of Jesus stating something which was misunderstood (Nicodemus thinking he needed to be literally reborn, for example).
If a person is in Christ they are a new thing, a new category which the world does not fully grasp. In the Roman world, it was very hard to understand why these Christians were willing to give up their lives for a belief in a crucified God. They did not understand why faith in Jesus brought men and women, slaves and free together as a new family, and they certainly could not understand why Christians would help the poor, care for the needs of the orphan or widow, help people who were sick with dangerous diseases, etc.
Even though we are children of God, we have cannot fully understand what this means (3:2). Continuing the metaphor of a parent and child, it is possible a child has no idea the exact wealth of their parent. Imagine a child who has a famous and wealthy father, but they do not realize who wealthy until they are an adult. As Karen Jobes says, “the full effect of being a child of the Father with eternal life is not something anyone can comprehend now” (1, 2, & 3 John, 142).
The one who has this hope purifies themselves because God himself is pure (3:3). This “hope” refers to the love god has lavished on us, but possibly as far back as the hope of confidence on the day of judgment.
To be pure has the sense of both ritual and moral purity. The verb (ἁγνίζω) is used for Judeans who purify themselves as they go up to Jerusalem to worship at Passover (John 11:55). In Acts 21:24 Paul goes up to the Temple to assist some men who were completing a Nazarite vow, they “purify themselves.” The verb is used as a metaphor in the New Testament, James 4:8 refers to purifying one’s heart and 1 Peter 1:22 refers to purifying one’s soul.
It may be a coincidence, but Paul does not use this language when describing sanctification, possibly because it is a very Jewish metaphor; the one who is purifying themselves is getting ready to worship God at the Temple. They are drawing near to the presence of God and therefore take the necessary steps to be ceremonially clean.
The child of God purifies themselves. Being born of God was passive, we do nothing to be born again, but John says there is some responsibility on the part of the believers to purify themselves.
Developing the Father and child metaphor just a bit more, if the Father is has some characteristic, then his child likely has the same attribute but the child will needs to make some effort to develop that attribute be more like the father. In this case, God is perfect purity, the child will not reach that same kind of purity, but that is what the goal ought to be.