There are really two parts to your question - Why did Islam take over Persia so quickly, and why were the people in political power when Abbasid authority broke down Muslim and not Zoroastrian. I'll answer just the first of these, since I'm not so knowledgeable on what happened to the Zoroastrian nobility after the Islamic conquest.

One of the, if not the main theorist of conversion to Islam is Richard Bulliet, who wrote the book Conversion to Islam in the Medieval Period. Harvard University Press, 1979. I won't go into his methodology, but he concluded that about 50% of the population had converted to Islam by the time of the appearance of native Iranian dynasties. You can see his graph here.

 

As for the why: Islam: The View from the Edge. Columbia University Press, 1994; and Cotton, Climate, and Camels in Early Islamic Iran: A Moment in World History. Columbia University Press, 2009, are two more books where he addresses the history of conversion in the early Islamic period, among other things. In the first book, he links conversion to the extraordinary and rapid urbanization which the Muslim world (but particularly Iran) experienced after the Islamic conquest. The conquering Muslim armies had established garrisons in each city, around which the Muslim community congregated. As Iranian cities were initially so small, this effectively meant their Islamization. Incoming migrants then came to adopt the religion of the city they were migrating to, and became Muslims. In the second, he argues (plausibly, but on less than secure evidence) that Iran experienced a cotton boom in the ninth and tenth centuries. Muslim landowners expanded over the countryside, bringing wasteland into production to produce cotton for refinement and export. This cotton boom affected conversion by encouraging those newly established villages to become Muslim, and by providing the impetus for urbanization (with its aforementioned impact on conversion), as economic opportunity in the cotton-market increased.