Did he have hard evidence to back up any of his claims or was it pure propaganda/conspiracy? Why was his rhetoric so convincing to people?
The claim of the stab in the back is that Germany was "undefeated" in November 1918 and there were not any Allied soldiers in Germany, but a rebellion at home organized by the SPD abolished the Imperial government, proclaimed a republic, and forced Germany to agree to an armistice. Coupled with Hitler's view that left wing politics = sinister Jewish conspiracy, and the argument becomes "Left wing radicals puppeted by Jews betrayed our country and capitulated to the enemy while we were still fighting."
Hitler did not invent this argument. In the early days it was mostly associated with Erich Ludendorff, who had been second in command of the German military and was eager to shove blame for Germany's defeat on someone outside of the top ranks of the German military.
The claim is ridiculous. The German military was soundly beaten by the end of summer 1918 and the November 1918 rebellion was aiming to set up a government the Allies would be willing to negotiate with. The war was already lost: Austria-Hungary had made peace and disintegrated three weeks before, the Ottomans surrendered two weeks before, and Germany was losing ground and in no position to keep fighting.