I, even I, am the one who wipes out your transgressions for My own sake, And I will not remember your sins. [Isaiah 43:25]
This verse contains quite a few things to note. It teaches us that only God can forgive, and the reason we have His forgiveness is “for his own sake, for his mercies’-sake, his promise-sake, and especially for his Son’s sake, and that he may himself be glorified in it” (quoted from Matthew Henry’s commentary). Then after God forgives us, he will remember those sins no more, as though a debt is cancelled.
Furthermore if we read this verse in context, the preceding verses list the sins of the people eg not worshipping God with burnt offerings, and burdening God with sins. Yet instead of punishing the people, God said He would forgive them. Such is His grace and mercy!