How did the illustration 'The Bradys and the Chinese Dwarf' help to foster anti-Chinese sentiment in the last half of the 19th...

First of all, this illustration could not have fostered
anti-Chinese sentiment in the last half of the 19th century since it's dated 1907. 
However, nitpicking aside, it does show the sorts of fears that led to persecution of
the Chinese during the time that you mention.

During the
late 1800s, many white Americans feared that the Chinese would come and take jobs away
from Americans.  They also feared what they saw as "inscrutable" Orientals who were
prone to creating criminal gangs.  You can see this second fear in the illustration. 
The Chinese are going to burn these Americans alive, probably because the Americans have
uncovered a Chinese criminal organization.

One of the
reasons Americans disliked Chinese was because it was feared that they and their gangs
would take over parts of the US economy where Chinese were plentiful.  This is shown
quite clearly in the illustration.