Mussolini was the leader of Italy during WW II. Italy was part of the Axis powers which included Germany and Japan. When he resigned, the Jews and people supporting the Allies were hopeful that this would weaken the Axis and shorten the war. With the Allies invading Italy, they now had a foothold in Europe and would continue to push north toward Germany and its defeat. With Germany being threatened from their south, perhaps they would ease up on the conquering of other European nations, lessening their military presence. More Jews would be able to escape or go into hiding. With the British bombing Germany regularly, and pressure from the south of Europe, surely the war would end sooner.
Mussolini's resignation is a hopeful sign because it demonstated the power of the Allied forces attacking Europe. Mussolini was the Facist Leader of Italy. If he could be forced to resign, it gave hope that Hitler might one day be forced to resign and the persecution of the Jews would be over.