This question really needs to be answered separately for different European countries that became right-wing dictatorships between 1918 and 1939.
A few general points. Fascist (I will use the term loosely, strictly speaking it is specific to Italy: Germany was Nazi) ideas drew on developments in nineteenth century thought that stressed the importance of nationalism, developed ideas of inherited intelligence and interpreted the new science of genetics, and of evolution, in racist ways. To go into detail about this you would have to consider the work of philsophers like Hegel and Nietsche (neither were fascists, but their ideas influenced fascism) and of nineteent centruy scientists in foelds such as archaeology and philology (study of the origins of language).
The chaos caused by the First World War undoubtedly helped people look for new political systems. On the one hand the looked to communism, and fascism was a reaction to this by people who believed that human society was inevitably hierarchically structured. The needs for an alternative to communism made it attractive for financiers and industrialists to give fascists essential right-wing support. In Germany, willingness of right-wing parties, including the Nazis, to repudiate Versailles, and hence to rearm, was a major factor. However, the rise of Fascism cannot be ex[plained entirely in relation to the First World War because of the rise to power of Franco in Spain, which had been neutral (although it had been humiliated by the loss of its colonies in war between Spain and the United States in 1898).
The chronic economic depression of the years 1918 - 1940, and in particular of the great slump following 1929, appeared largely due the free market cultivated by liberal democracies, specifically the United States and United Kingdom, between the wars. However, an expanation along these lines needs to take into account that the Weimar republic of 1920s Germany was to a considerable extent run on democratic socialist principles as the Social Democrats were the largest party in the coalition governments. One might also ask why Thatcher's destruction of the UK economy between 1979 and 1983, when unemployment rose to 3 million, did not lead to similar political extremeism.
Issues relating to ideology, economics and the legacy of past conflicts are clearly important in explaining the rise of fascism. However, a short anwer is likely to be oversimplistic about them This is very dangerous. In recent years, politicians have got away with some very undesirable totalitarian policies. One reason is that if prewar precedents are pointed out the objector is too often criticised and ignored for labelling people as "fascist" instead of being listened to serious
