Short answer: Because Iran was already an ancient and powerful country with long history of running an empire and huge cultural influence in the region.
Richard Frye has written about this: "A complete model of imperial rule was thus presented to the Arabs by the Persian realm" ( Richard N. Frye, The Golden Age of Persia, London: Phoenix Press, 2000)
Not sure what you mean by "Arabized" but in fact Iran did absorb quite a bit of "Arab" culture too including many loan words in their vocabulary (Persian or Farsi only re-emerged around 1000 AD) but it also greatly impacted the invaders too. This has been the pattern of Iran whenever it is occupied by others: the others end up adopting Persian ways and themselves become Persianized (including the Arabs, Greeks, Mongols and the Timurids etc.)
By the time of the defeat by Muslims in 637 the Zoroastrian religion had become very oppressive and rigid. However Iranians continued to maintain many of their preexisting religious and cultural values even though they adopted Islam which was more egalitarian. On the other hand the Beduins who had no history of running an empire had to rely on the Persians as bureaucrats with long experience in such matters. Even the Prophet Mohammad credited the Persians: "Even if faith (knowledge) were near the Pleiades, a man from amongst these would surely find it." http://theonlyquran.com/hadith/sahih-muslim/?volume=31&chapter=59
By the time the Abbasids had defeated the Ummayads in 750AD, Persian influence had grown to such a degree in the Caliphate that the Arabs had adopted the administrative practice of having a "Vezir" or Prime Minister as the chief executive, and since the Abbasids were based in Khorasan, today a province of Iran, it was inevitable that they would absorb Persian ways. They adopted the Persian practice of maintaining a large standing army, their capital was transfered from the Arab Damascus to Baghdad (close to the formerly Sassanid and Parthian capital of Ctesiphon, near where the Sassanids had established a famous medical univeristy https://en.wikipedia.org/wiki/Academy_of_Gondishapur )and the had revived and adopted the pre-islamic Sassanid administrative practices and had translated Sassanid books. An Iranian family, the Barmekids (originally Buddhist) were the chief advisors and adminsitrative officers of the Abbasid caliphate for a generations https://en.wikipedia.org/wiki/Barmakids