In this sentence the "writer" is making sure that their readers understand that they are "rich" by confirming a second time "that they are" rich. By doing this the writer is baicly letting you know that they are very rich.
They are poor, that they are. = very poor
They are hungry, that they are. = Very hungry
They are learning English, that they are. = Really putting a lot of effort into learning it.
I hope this helps,