How does the atonement work? What did it 'do'?
What was put on Christ was essentially an eternity of morally imperfect intentions. This is what sin is; holding bad intentions towards others. Sin isn't like a crime where you go to prison for a specific amount of time, it's basically just bad intentions of the heart (1 John 3:15; Matt 5:27-28).
That's why sin goes away when you repent. It goes away because one's intentions are no longer bad if one changes one's behaviour for good.
But repentance doesn't stop a person from continually sinning and repenting forever, in a neverending cycle of sin and forgiveness. God doesn't want us to live in the Kingdom of God and be continuously sinning and repenting, because that's a bit of a difficult situation. It's not good enough for God, not up to God's standards, to be permanently involved in a cycle of sinning and repenting forever and ever. God wanted to deal with all our sin once and for all.
That is the difference between the atonement and just asking God for forgiveness - the atonement dealt with sin once and for all, whereas asking for forgiveness deals with sin until we sin and ask for forgiveness again.
God prevented a situation of people in the Kingdom of God constantly sinning and repenting, which we do in this life, by putting all of our wrong intentions, including all the ones that we would have had over an eternity with God (in every possible future in the Kingdom with Him) onto Christ.
Because intentions come from the world of Mind, they need to be transferred to something in the world of Mind to get rid of them. They can't be transferred to a stone, for example. So our bad intentions needed to be transferred to a Mind. We could also just have repented and asked God for forgiveness, but as we saw a situation of continual sinning and repentance isn't good enough for God's Kingdom.
The mind that God planned to transfer our wrong intentions to had to be a perfect mind, because this process should be two-way. We need to get something back for our wrong intentions, otherwise we'd be replacing our morally wrong thoughts with no thoughts at all (a complete absence of thought), or the thoughts of another, equally flawed person. That would not be a huge improvement for us either way.
We need to get back perfect intentions forever, to allow us to meet God's standards and thus be in heaven. That's why the sacrifice needed to be Mind, and a morally perfect mind, rather than a) a stone, b) losing evil thoughts with nothing to replace them with, c) a flawed mind, or d) just asking God for forgiveness again and again.
No human was found worthy (Rev 5:2-5) to be the 'transferral person', because no human is capable of having perfect intentions all the time, so any transfer of bad intentions from us to him/her would have been pointless. We'd just have gotten flawed intentions back.
So God had to become a human, a morally perfect human, to be the mind undergoing the transfer (i.e. sacrifice).
Once all of our wrong intentions were transferred to Christ, then Christ had to suffer an incredibly painful separation from God, as would anyone who took on that degree of evil intention (literally an eternity of evil intentions over an infinity of hypothetical situations that we could have sinned in, per mind, for billions and billions of minds).
Because Jesus is, in fact, a morally perfect person, He did not bear our evil intentions forever. After a certain amount of time, the wrong intentions placed upon Jesus simply 'evaporated' somehow. Meanwhile, all of our wrong intentions over an infinity of hypothetical situations had been swapped for Jesus' perfect intentions. So Christ within Christians does everything in the areas where we used to sin, making us meet God's standards once and for all with no good works necessary.
This allows us to be in the Kingdom of God without having to constantly repent and get forgiveness, because in this way God has prevented us from having morally bad thoughts, as all of our morally flawed thoughts (but one, see end) were given to Jesus. This is the only way that God could give us a new heart (Eze 11:19) because He needed a mind-mind transfer to get rid of our flawed intentions, and to get good intentions back that mind needed to be perfect.
By being 'atoned for' in this way, for the first time people will actually be able to have their soul (which is made in God's image) connected to God as opposed to disconnected from God. This will allow Christians to be as happy as God is, a happiness that no sinner has ever known and literally cannot imagine (1 Cor 2:9).
Christians remain very imperfect individuals because this process hasn't fully occurred yet (it fully occurs on the day of redemption - Eph 4:30). God wants to use our sufferings and struggles on earth to 'refine us as silver is refined' (Zec 13:9). Through our struggles with sin, we learn to rely on God, so God has delayed the full effect of the atonement until after we die. Through learning to rely on God we won't be tempted to leave God when we're in the Kingdom of God. Someone who is hesitant about entering the Kingdom of God and would want to leave it is not fit for it (Luke 9:62). So it's important that we learn not to want or do that.