"This is how God showed his love among us: He sent his one and only Son into the world that we might live through him. This is love: not that we loved God, but that he loved us and sent his Son as an atoning sacrifice for our sins." NIV translation
John reminds us of the example of this sacrificial love in God sending His Son for us. If you have children, you can probably understand more of what this means as you can imagine giving up your child to save someone that hates you. That is what God did for us and that is what John means by atoning sacrifice.
"Dear friends, since God so loved us, we also ought to love one another. No one has ever seen God; but if we love one another, God lives in us and his love is made complete in us." NIV translation
Because of the love that God demonstrated for us in sending His Son to the cross, we should love others as He does. He gave us the Holy Spirit, which is God living in us. Because God now lives in us and He is love, we have the ability to show love to others. Love is complete when it is given and so, when we show that love to others, God's love that lives in us is made complete.
"We know that we live in him and he in us, because he has given us of his Spirit." NIV translation
Got the Spirit? If you do, then you are a child of God and you can be confident in that fact. Paul told us, in Ephesians 1:14, the same thing and even went on to say that the Spirit is a down payment of what we are going to have for eternity.
"And we have seen and testify that the Father has sent his Son to be the savior of the world." NIV translation
John was there throughout the time that Jesus walked this earth and also saw Him after His resurrection. The "we" refers to us as well for, when we accept Jesus Christ, God gives us the wisdom and the Spirit to know and testify about His grace through Jesus Christ.
"If anyone acknowledges that Jesus is the Son of God, God lives in him and he in God." NIV translation
John tells us again of the paternity test and how we can tell whether someone is a child of God or not. It is quite simple in that they must believe that Jesus is the Son of God and not just a good man or a just a prophet. This statement has offended many people throughout history and even more today but we cannot compromise on this fact.