Dickens probably meant that just as The Old Curiosity Shop had gone, so bit by bit people forgot about Nell, the shop and everything that had happened there. All these things had been very important to Kit, but as the years passed he started to forget the details, and after many years
all the people in the story were no longer alive and so they were forgotten. When Dickens said that ‘Things pass away ... just like a story that has been told’ he was also referring to the story of The Old Curiosity Shop, and so he meant that the story was now nished.