Great question. There's a lot to unpack here but much of it boils down to the difference between what it means for us to forgive someone personally versus someone being pardoned legally. The short answer is that Jesus died a self-sacrificial death on the cross to pay the penalty for our sins.
To back up for a minute though and set some context: God is the only one who can forgive sins, because it is first and foremost Him that sins are committed against in an ultimate sense (since it His law is the highest authority that they break). God, in His Mercy is willing to overlook our sins and restore our relationship with Him, but Mercy is not God's only attribute. He is also Just; and justice requires that a penalty or debt must be paid. If all God did was wave His hand at sin and the scriptural penalty He has assigned to committing it then He would invalidate His law and there would be no justice at all. Even in our fallen state, as humans made in the image of God, we feel outrage when evil doers escape justice and recognize a fundamental "wrongness" of a situation in which a heinous act is committed and yet no one is made to 'pay for it.'
So God cannot simply ignore our sin, but He can't ask us to pay for it ourselves either if His goal is to reunite us with Him one day because of the scale of the penalty. Everyone instinctively knows that the punishment should scale with the offense (a murderer should not get a slap on the wrist and a child who says a naughty word shouldn't be whipped). So, given the extent to which our sensibilities are offended when someone else unwittingly cuts us off in traffic so that we must "endure" the minor inconvenience of moving our foot off the accelerator to the break to slow down, what must the penalty be for someone who has repeatedly (and often gleefully) offended the infinitely Holy God, creator of heaven and earth, who holds their very breath in His hand, and actively causes them to continue to exist moment by moment? Clearly, any objective review of the situation would quickly show us that there is no limit to the penalty incurred. There is no point at which a mere man can or will have paid enough for reckless depravity.
What can God do then? Given that sin can't be ignored and we can't afford to pay the penalty for it and live, how can the relationship be restored? The answer could be the sacrifice of a another life than our's, but where should this other life come from? God could demand it from an angel maybe or another man, but that would only make a mockery of justice. Again, this is something we all know inherently: that the innocent ought not to be punished with the guilty and Abraham even brings just such an objection up when pleading with the Lord for Sodom in Genesis.
In conclusion, we see that the ONLY way for God to restore our relationship with Him without violating His own nature of Justice or Mercy is if He were to pay the penalty for us with His own life, and that's precisely what He did by incarnating as a man to live and die in our place. This is the heart of the Gospel: John 3:16, "For God so loved the world [that's us!], that He gave His only begotten Son [Jesus Christ, the co-equal, co-eternal, 2nd person of the Godhead; separate in personhood and yet of a common Godhood with the Spirit and the Father], that whosoever believeth in Him should not perish, but have everlasting life."