That is a really broad question and "youth" is also a rather broad term, so I can only give a broad answer concerning Germany:

After WWI and the experience of violent revolution, hyperinflation and the difficult early years of the Weimar Republic, there in essence was a generation growing up that was influenced by the violent rhetoric and aversion to compromise that was so essential to the war and the subsequent years. In short, through the school and university system, there was not only disillusion with the workings of democracy but the widespread fear of a fundamental threat by an enemy in the form of communism that sought violence. And thus the answer had to be violence. Especially those who had grown up during WWI but never experienced the horrors of the battlefield were enthralled by the violent rhetoric of the time but unable to reconcile it with the actual realities of said violence. Fostered by people such as Ernst Jünger, a whole generation was brought up in Germany in the spirit of the noble fighter giving his life for a greater cause against an enemy so foreign and vile, no middle ground was possible.

These people were essentially what made up the Nazi vanguard in several sectors such as the SS and its Reich Security Main Office. Michael Wildt found that of those active in the RSHA and the Einsatzgruppen, most were born at some point between 1901 and 1909 and thus knowingly experienced WWI but never participated in it.

Furthermore, Nazism offered young people the feeling of being part of something greater and being able to help build a new world order through the Volksgemeinschaft. This is a powerful motivator that can not be underestimated in its appeal.