How did the Versailles Treaty and Economic Depression allow the rise of totalitarian leaders and dictators?

The Versailles Treaty put all the blame on the Germans for starting World War I and all the damage it caused. Germany was forced to pay billions in reparations to the victors, which caused severe inflation in Germany due to all the money they had to print. The Germans lost the war, and were very resentful over the reparations and demilitarization, and were hungry for revenge and to make a come back. Many Germans wanted another war since they thought victory has been stolen from them by backstabbers, especially Jews, within their country. Hitler promised all this, and also scapegoated the Jews for losing the war and the economic hardships of most Germans after the war. Hitler also convinced the Germans that they were a "Master Race", and the socialists and inferior races living within Germany had weakened her. The Germans as well as the Italians were looking for strong leaders to restore order and make them feel proud again. Fascism appealed to this.

Arnold K has expertly answered your question but I would like to add one other point: the Allies - Britain, America, France and Russia all ganged up against Germany. They stole great swathes of land. Alsace-Lorraine went to France, the Malmedy area to Belgium, and most of Posen and West Prussia to Poland. Danzig was made into a free state. And to add further humiliation, the Allies would occupy the Rhineland for another fifteen years. After the Treaty was signed, Hitler realized that Germany's population would not have enough room left to expand. He called this 'Lebensraum' meaning 'living-space'. That's why he invaded countries to the East - to secure enough land for Germany's future generations.

desperate times call for desperate measures and the losers of WWI were some of the most desperate, although the economic situation was bad in most of the world