I'm not familiar with the rise of fascism in Italy, but I can speak to Germany. Prior to WWI, Germany was one of the greatest countries in Europe. They had the largest economy, the largest industrial base, and were a hub for science and learning. The German Empire seemed a major success, having unified the disparate German states (except Austria), under its rule and easily defeated France in the 1870s. WWI changed all that. Germany suffered a humiliating defeat, and for most Germans an inexpelicable one. After the war Germany spiraled into chaos. While most Germans, and even many international observers, believed that various countries carried blame for WWI, the Treaty of Versaille required Germany to symbolically take all the blame. The treaty also required Germany to make expensive and humiliating reparations payments to the victors. The German military, which had been one of the largest and most powerful in Europe, was reduced to a fraction of its size by the treaty. Economically, Germany was in shambles. The economy collapsed post-war. While other countries experienced a boom in the 20s, Germany was mired in a depression. Hyper inflation wiped out savings and many places suffered from scarcity. Politically, the country was also a shambles. The Weimar government set up after the war was democratic, but poltiically weak. There had been a failed attempt at a communist revolution right after the war. Throughout the 20s, various political factions brawled in the streets and carried out political assassinations against one another. It was chaos.
The Nazis promised to end all this. WIthout meaning to make an analogy with modern politics, they promised to make Germany great again. They said that they would rein in the political chaos in Germany (which they had been a leading contributor to) and that they would restore the economic health and international prestige of the country. Many Germans were willing to listen. Remember that Germany and most other European states didn't have an extensive history and tradition of democracy. The German Empire had been a constitutional monarchy, with a wide franchise and an elected parliament, but it had only existed for fifty or sixty years. This wasn't like the US, which by the 1930s had been a democracy for over a century and a half. The idea that maybe democracy has problems wouldn't have seemed as crazy to Germans. And keep in mind that the Nazis weren't out there campaigning in the late 20s and early 30s on the idea of a dictatorship. That really only came to fruitiion after Hitler was named chancellor in 1933. The Nazis were the largest party in the Reichstag, the German parliament, at the time (although not an actual majority) and German President Paul von Hindenburg eventually asked Hitler to form a governing coalition and rule as Chancellor. Well, a few weeks after Hitler assumes power a fire broke out in the Reichstag building, severely damaging it. The German authorities arrested a mentally ill Dutch Communist for the crime. The Nazis used the Reichstag Fire to get passed a new set of laws giving the Chancellor more power and restricting civil liberties, especially for Communists. Hitler used these news powers to stamp out resistance. In the next election, which was carried out under a rigged system, the Nazis won an overwhelming majority and then passed new laws which effectively ended German democracy. At the time, most Germans probably accepted these steps as necessary ones to deal with political unrest. And Nazi support grew in the coming years as they seemed to make good on their promises. The economy was improving (it had begun to improve in the early 30s but the Nazis ended up reaping a political benefit from it). The Nazis then worked hard to restore German prestige, building the military back up and expanding German territory. Prior to 1939 there would have been a lot for your average German to like, as long as you didn't care about democracy and were fine with Jews and Communists being persecuted. And after 1939, the Nazis lead the Germans to a series of amazing military victories. They conquered Poland in about a month with few German casualties. Even more impressive was the defeat of France. In WWI, they had spent four years fighting, and losing to, the British and French. In 1940, the Germans utterly defeated the French and British, forcing them to feel France. They conquered half of the country in just a month and turned the other half into a German puppet state. Again, as long as you didn't care about the loss of political freedoms there was a lot to celebrate.