What is the purpose of this illustration?Illistration:...

The purpose of the illustration that you linked to is to
sell books.  As you can see, the illustration is from the cover of a dime novel (cheap,
sensationalistic fiction as opposed to "real" literature).  A publishing company will
only put something on the cover of a book if they think the illustration will help to
attract people and make them want to buy the book.

So the
more important question here is why the publisher thought that that illustration would
help the book sell.  The answer there is that the illustration taps into the sort of
fears that people had back in those days.  In the present, we have TV shows and movies
with trailers and ads that play on our current fears (terrorism, crime, etc).  So it was
in those days.  There was a fear of Chinese immigrants and their supposedly lawless
ways.  Because of this, a book that featured Chinese criminals (and which showed them on
the cover) would be likely to sell, just as a show about someone fighting terrorism is
likely to do well today.