The 'frontispiece' is the illustration you find on some books as you first open it up.
It's usually one of the first pages you find on the left and is opposite the title page.
Here's some images that show it.
A 'period piece' is usually some art work (in this case a book) that either belongs to a particular time period in the past, or it captures or highlights aspects of that particular historical period of time.
So, in this example Mrs Driffield is referring to the illustration or art work on the opening left page of some book. She is describing it as a 'period piece', as in some way, this artwork is highlights/shows or is typical of this period of time.