What did Jesus quote Psalm 22: "My God, My God, why have you forsaken me?"

In my opinion God didn't turn his back but merely his gaze. Like any father asked to watch his son endure torture or pain God had to look away so that he would not stop what was always supposed to happen. It was written that Jesus would be born of a virgin, which He was. It was written that Jesus would be betrayed. Which he was. It was written that Jesus would then die for the sins of the world to be forgiven and that was done so that he didn't have to go back on his promise (with a rainbow) to never do what he did in the days of Noah again. Jesus died so we didn't have to. Our sins were all forgiven. We should however thank him for this gift by trying our hardest not to sin. Just cause we are forgiven doesn't give us free range to sin as we please.

In Mark's Gospel, the last 24 hours in the life of Jesus form a unique chiastic structure, with eight periods of just three hours each:

A The celebration of the Passover Feast, which becomes the Last Supper, beginning "when it was evening" (Mark 14:17), which was 6 pm by first-century Jewish reckoning. The Passover meal typically took three hours and was concluded by singing a hymn.

B When was about 9 p.m. Mark then has Jesus and the disciples go to the Garden of Gethsemane, where Jesus went to pray, asking that, if it be possible, his Father take this cup (his destiny to be crucified) away. He returned three times and each time found the disciples asleep, measuring three hours of time by asking, "Could you not sleep one hour?".

C The betrayal by Judas, the darkest deed in human history, came next, occurring at the stroke of midnight. This will be reflected by the darkness at midday.

D At 3:00 a.m., Jesus was led away for a trial before the high priest and other senior priests and elders. We know the time of the first trial because Peter's threefold denial of Jesus followed, once each hour until the c**k crowed, marking the watch between 3 a.m. and 6 a.m., known as c**kcrow.

E When it was 6 o'clock, "As soon as it was morning", Jesus was led by the chief priests, scribes and elders for trial by Pontius Pilate.

D' At 9 o'clock: "It was the third hour when they crucified him."

C' When "the sixth hour had come" (12 noon), darkness covered the whole earth, reflecting the betrayal at 12 midnight.

B' The three hours of darkness, until 3 p.m. mirror the agony in the Garden of Geths*m*ne. Jesus last words, "My God. My God, why hast thou forsaken me?" reflect the recognition that his prayer in the Garden has not been answered.

A' Joseph of Arimathea then asked Pilate for the body of Jesus, so that he could be buried before the Sabbath began at 6 p.m., when the sun went down.

In this structure, event B' represents the moment when Jesus said, “My God, My God, why have You forsaken Me?” and matches event B in the Garden of Geths*m*ne, where Jesus asked that, if it be possible, his Father take this cup (his destiny to be crucified) away:

Mark 14:36: And he said, Abba, Father, all things are possible unto thee; take away this cup from me: nevertheless not what I will, but what thou wilt.

Jesus felt forsaken because, in his prayer in the Garden, he has asked for this cup to be taken away from him, although he had also said that if it be God's will, he would submit obediently.

There is no particular correlation between Psalm 22:1 and Mark 14:36, other than that the words suited the evangelist's message for Jesus. The author of Matthew's Gospel recognised that these words were from Psalm 22 and retained them when copying from Mark. However, the authors of Luke and John substituted different words that suited the different theological requirements of each evangelist:

Luke 23:46: And when Jesus had cried with a loud voice, he said, Father, into thy hands I commend my spirit: and having said thus, he gave up the ghost.

John 19:30: When Jesus therefore had received the vinegar, he said, It is finished: and he bowed his head, and gave up the ghost.

Jesus was really suffering. He probably could have used miracles to not feel any pain, but then the whole sacrifice would just have been a fake one. To be a real sacrifice, feeling real human pain, both physical and psychological, was indeed necessary in my opinion. Can there be any bigger emotional pain than feeling that God has forsaken you?

Take a look at this article which I feel explains one view pretty well.

"It is possible that at some moment on the cross, when Jesus became sin on our behalf, that God the Father, in a sense, turned His back upon the Son. It says in Hab. 1:13 that God is too pure to look upon evil. Therefore, it is possible that when Jesus bore our sins in His body on the cross (1 Pet. 2:24), that the Father, spiritually, turned away. At that time, the Son may have cried out."

http://carm.org/questions/about-jesus/why-did-jesus-cry-out-my-god-my-god-why-have-you-forsaken-me

The quick answer, the condensed one that I hear in homilies at Mass around Holy Week is that Jesus quotes this scripture to evoke the Psalm, not just the part of the Psalm where David feels forsaken, but the part where God is glorified.

For the same reason so many other things from the Old Testament are mentioned in the New Testament: they're quotations. This particular one was a reference to the beginning of the 22nd psalm, and it's quite instructive to look at the psalm in its entirety. Of particular interest is verse 18, which had literally happened to him just moments ago. Jesus's invocation of the psalm gives a good look into his mindset and his feelings at the time.