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