As has been said here, Christ does not suffer on the cross in order to appease divine justice or honor. The idea of "penal substitution" i.e., that Christ is punished in place of humanity, is NOT an Orthodox teaching.
In order to begin to understand the cross as the Church does (I say "begin" because of the depth and beauty of it. I believe most of us have only "begun" to understand), you must first begin to understand the Orthodox view on salvation. Other traditions think of salvation as going to heaven instead of hell. However, Orthodoxy sees these places as states of being more than actual locations. Many Orthodox believe that at the end of time, all will experience the full love of God. Those who accept it experience warmth, joy and love. Those who choose to reject it experience God's love as pain and sorrow. As the Scriptures say, "our God is a consuming fire."
So, salvation for Orthodox is uniting yourself to Christ, to experience the "renewing of your mind" as St. Paul says, "mind" in this case meaning that operating principle that makes someone human ("nous" in Greek). By being baptized and chrismated, by partaking of the Eucharist, confessing our sins, fasting, praying, etc. we are conforming ourselves to Christ-likeness. This is what St. Peter means when he speaks of becoming "partakers of the divine nature."
This presents us with a problem. How can human nature partake of divine nature? They are entirely different. The answer is Christ. The second person of the Godhead, the Son, becomes incarnate of a woman. God takes on human flesh, uniting human nature and divine nature in Himself. This is why it is so important that Christ is FULLY man and FULLY God. He must take on every aspect of human nature. If he does not, humanity is not saved.
So, Christ saves humanity be uniting our natures. He makes it possible for humans to be partakers of divinity. Unfortunately, because humanity is exposed to corruption and death, Christ is not yet done. He must unite Himself to every aspect of humanity. He suffers hunger and pain, he is tempted to sin, and lastly...he must die. So, he does, on a cross. He offers himself, sacrifices himself, and takes on death itself. However, He is God. He is the author of life. Life itself, the essence of life, cannot die! Christ unites God to death, and that cannot happen. Because of this, death is destroyed. Every Sunday at Matins we sing, "death has been slain, so we are given life." Christ shatters death and provides the way out. Death is not longer permanent. Christ proves this by rising from the dead on the third day, giving us the same promise of resurrection in the last day.
So, Christ is the Lamb of God who takes away the sin of the world. But salvation is more than "just" being forgiven. Salvation is becoming indwelt with the Holy Spirit, becoming partakers of the divine nature by the renewing of our minds. Being in eternal communion with the limitless God of the universe. THAT is the high calling of the Orthodox Christian. It is much more than simply being forgiven.
I know my reply is very long, but I hope all that helps.