To atone is to make amends for, or to compensate.
In the context of the law, people who commit crimes or civil wrongs are often given punishments or ordered to pay damages in order to atone for their actions. Damages, jail sentences and other punishments are designed to correspond proportionally to the action.
Often, defendants in criminal cases plead guilty. Doing this means that they are acknowledging and admitting that they committed the crime. Defendants who plead this way do so with the knowledge that they will be forced to atone for their crimes by paying fines, going to jail or facing other consequences.
This can also happen in civil cases, where defendants admit that they caused the civil wrong, and express a desire to atone for their wrong(s) by paying damages.