The emphasis in these three verses is on the last verse quoted from Jeremiah 31. The writer has already quoted the same passage in Hebrews 8:10-12, and now quotes it again with some variation of word order and without including all that was included the first time.
This is good news beyond all expectation. God is said to remember no more our past sins, but of course, God cannot actually forget anything that has happened. What this means is that he will never again bring those sins into consideration in any of his dealings with us, his interactions with us; they will never again have any influence on how he treats us. In human relationships, it may well be that there has been a sharp quarrel in the past, and it seems to have been forgotten about. However, some new dispute between the parties may cause one of them to dredge up this former disagreement from the past, and use it to add weight to their present quarrel. God promises that he will never do this. ‘I will remember no more.’ Our sin is cast into the sea of his deliberate forgetfulness. We need this assurance, because God’s hatred of sin is so great that we might feel that he could not help remembering it on some future occasion, and certainly there is enough in our past sin to condemn us. But no, by his deliberate choice, he will not remember it; we are safe.
The Greek starts by positioning us in Jeremiah at the words already quoted in Hebrews 8:10 and directs our attention to what follows with the phrase, ‘After that he had said before’ – ‘After that he had said before, This is the covenant that I will make with them after those days’. After this, what? The writer to the Hebrews expects his readers to know the quotation well enough to be aware he is passing over some material in Jeremiah in order to come to the last part, to which he wants to draw our attention. In English we would probably add words such as, ‘Then he says’ – ‘Then he says, And their sins and their iniquities will I remember no more.’