Constant propaganda, xenophobia and nationalism can get hundreds of thousands if not millions to mourn a dictator who killed millions of their fellow citizens in the name of democracy, national glory and protection against the USSR's great satan.
Now toss the same kind of propaganda, xenophobia and nationalism and make it two-fold: Not only is the loss of the Iranian empire one of the greatest disasters of the 1st millennium and outsiders to not be trusted, but as Shi'ites they are the best people of the best people.
(And as for crowds, it's like the grassroots Soviet protests: You get not only time off from work for a protest, but you get to show lealty to the regime and that helps both your job prospects and gets your kids better recommendations for college. In Soviet times, the banners wouldn't be produced the day of the protest, you'd get time on the job the day before to make them and work-time to organize and figure out how to spontaneously protest.)