Please note that it doesn't say that agape love is one of God's attributes. No, the Bible states that God IS agape! He is not agape plus....
What, then is "agape"?
The best quick description can be found in 1 Cor 13:5. Let's look:
"Love...is not self-seeking". So the opposite of self-seeking is a selfless love that lives only for others.
But a clearer picture can be seen from the love of Christ revealed at the cross.
When Christ became a man in the incarnation He had to give up not His divinity, but His divine prerogatives, in other words, the independent use of His divinity. Even His God-consciousness had to be given up. Jesus discovered He was God only by revelation. He was not God-conscious as a baby. He had to grow up in knowledge. He had to grow up in everything because He had given up the independent use of His divinity and was made in all things like unto us (Heb. 2:17).
Therefore, He was totally God-dependent all through His earthly ministry. John 5:30 says, “I can do nothing of myself.” John 6:57 says, “I live by the Father.” See also John 8:28 and John 14:10. All these texts state very clearly that Christ was totally God-dependent. Then read Rom 6:4; Acts 2:24, 32; Eph. 1:20. All of these texts clearly tell us that it was the Father who raised Christ from the dead. Keep these two things in mind: Christ was God-dependent, and He was dependent on the Father for the resurrection.
Do you know what the Father did on the cross? Christ cried out: “Father, Father why have you forsaken me?” What He meant is not “Why are you leaving me for three days?” but, “Why have You abandoned Me?” Do you know what that meant to Christ? It meant that the hope of the resurrection went with that abandonment.
As He hung on the cross experiencing God’s curse for our sins, Jesus had to make a choice. He could not save Himself and the world at the same time. And He did make the supreme choice. He chose to die eternally that you and I may live in His place. “But God demonstrated His love that while we were still sinners Christ died for us” (Romans 5:8).
In other words, the supreme sacrifice is that Jesus was willing to accept our curse and give us His life in exchange. It was not a question of saving Himself (hence self-seeking) and the human race. He could not do that. He had to make a choice between the human race and Himself. Do you know what Christ was saying on the cross? I hope you will never forget this; He was saying that He loves us more than Himself. That is God’s agape!