You need to consider the situation in its full context. Let's go by it passage by passage, fair warning I'm not a proper theologian and encourage others anons to correct me. I should not and do not replace a properly ordained priest.
lets start at John 8:2, as 8:1 is meaningless without prior passages.
>Early in the morning he came again to the temple. All the people came to him and he sat down and began to teach to him
two important things, first Jesus is doing this early in the morning, both the teacher and the students consider his teachings to be the most important thing one could be doing in the morning. Secondly it states all the people, not all the Jews. Which means there very well could have been Romans who wandered in half curious and got engaged in the wisdom of the Lord.
>The Scribes and the Pharisees brought a women who had been caught in adultery; and making her stand before all of them,
During his teachings the Scribes and Pharisees interrupt the Lord's teachings, imagine if during a sermon you stood up and shouted at the priest "Yeah well what about X?" Whatever point they had is made moot by their interruption and rudeness towards the Lord, by speaking out of turn they completely disrespect the teachings.
>They said to him, "Teacher, this women was caught in the very act of committing adultery
Addressing Christ as Teacher is dishonest of them, as they do not regard him as a teacher of any sort. This is a trap they lay for him as to mock him and what he teaches, and you'll see the trap in the next verse.
>Now in the law Moses commanded us to stone such women. What do you say?"
This is the trap. Our Lord has been preaching the virtues of forgiveness and mercy prior to this. Now the Pharisees think they have one up on him as the law of Moses is cut and dry, stone the women and offer no mercy to her. They wish for him to either break the law and have him arrested or go back on his teachings.
>They said this to test him, so that they might have some charge to bring against him. Jesus bent down and wrote with his finger on the ground.
And here John agrees with my statement, they want to have Jesus arrested. Do note that it is unknown what Jesus wrote into the ground but for whatever reason John thought it important to note that he did, perhaps a wiser anon can enlighten us.
>When they kept on questioning him, he straightened up and said to them "Let anyone among you who is without sin be the first to throw a stone at her."
This is how Jesus not only disarms the trap, but wraps it around the neck of the Pharisees. He has given them full permission to stone the women, but only on the condition that one among them be without sin themselves. The Lord, being the only one without sin, has told them to do it, so they can not even command him to do it.
>And once again he bent down and wrote on the ground
The focus John has on this writing points towards it being important, but I am not a wise enough man to figure out why.
>When they heard it, they went away, one by one, beginning with the elders: and Jesus was left alone with the women standing before him.
They see the trap that Jesus has expertly disarmed and wrapped around their neck, and admit defeat by withdrawing. The elders, being wiser with age, do this first but none of them are prideful enough to think they are without sin, and so all go away.
>Jesus straightened up and said to her, "Women, where are they? Has no one condemned you?"
With the Pharisees gone, technically nobody is accusing her anymore. With no accusers left there is nothing to stone her for. Though Jesus could change this at any moment by accusing her himself, and being without sin, he could cast the first stone. Here he is the ultimate Judge of not just her body but her soul.
>She said, "No one, sir." And Jesus said, "Neither do I condemn you. Go your way, and from now on do not sin again."
This is the most important of the entire passage, and the last one. With all of her accusers having left, Jesus can without breaking the law set her free. He does this, granting mercy and forgiveness. Yet this does not come without cost, as he gives her condition. She may go her way, but she is to sin no more. This is teaching that second chances are very much a Christian concept. Should she be caught again in Adultery he will not save her, for she was given a chance to learn. That is the meaning of this Chapter. "Let he who is without sin cast the first stone" is only a call to non-violence if you read it in isolation. In truth it is a call to give a second chance and offer forgiveness. Forgiveness is not given without condition, but it is given regardless.
As we all live in sin, we should not be quick to judge but instead offer forgiveness understanding that we too are sinful beings trying to be better, and to fall to sin doesn't need to be a permeant status. Think of how many Saints, in the days of ancients and the modern era never would have had a chance to do the Lord's work had they not been given a second chance to straighten themselves out. Only the Lord, with his infinite wisdom and knowledge, can truly know if a person is serious when they repent, because not even the repented can truly know until their death if they actually meant it.