These two questions are fairly interrelated, but they're also belong to quite distinct phases in the history of the Safavid dynasty! The "prominence" of Shiʿism in Iran was, at least initially, due to the religious preferences of the Safavids and their tribal (qizilbāsh) supporters rather than a demographic trend across the entirety of Iran. (In fact, to this day, Sunnism is practiced in the Iranian borderlands, especially in the west of the country.) So to answer your first question, we have to ask another: why, and when, did the Safavids choose to associate themselves with Shiʿism in the first place?

This is a difficult question to decisively answer, for a number of reasons. First of all, prior to the consolidation of the Safavid empire as a rival to the Ottomans and the subsequent creation of rigid confessional boundaries, confessional ambiguity had characterized the Islamic world since at least the Mongol conquest of Baghdad. While today we might think of the veneration of ʿAli and his descendants as a distinguishing characteristic of Shiʿism, it was commonly practiced in those times by figures who might otherwise be identified as conventional Sunnis. In addition to widespread ʿAlid loyalty, the period also saw the rise of a number of more extreme religious groups, the ghulāt, who held beliefs far outside the bounds of modern Sunnism and Shiʿism, e.g., in the transmigration of the soul. Many of these groups were also messianic in character, and the early history of the Safavid state was in part characterized by the monarch's negotiation of the messianic space that they established. Some of the important groups in this respect were the Mushaʿshaʿ in Khuzistan, the Nūrbakhshīs in a number of urban centers in Eastern and Central Iran, and the Safavids themselves, who had led a Sufi lodge centered in the Caucasian city of Ardabil for generations before Shah Ismaʿīl I's conquest of Iran in the early sixteenth century.

But here's where the second caveat comes in—our knowledge of the earliest generations of the Safavid order is rather fuzzy, and based on a scanty source base. Still, most historians believe that insofar as the labels "Sunni" and "Shiʿi" meant anything before the sixteenth century, the founder of the order—Shaykh Ṣafī al-dīn Isḥāq Ardabīlī—was essentially a Sunni, as was his son. Consensus breaks down around the third generation: more recent historians (e.g., Rudi Mathee) see Safī al-dīn's grandson Kh^(w)āja ʿAlī as the helmsman behind the turn to Shiʿism, while an older view (as expressed in Michel Mazzaoui's monograph on the origins of the dynasty) holds that the essential transition occurred after the death of Safī al-dīn's great-grandson and the temporary fragmentation of the order into two branches. Regardless, I suppose, all agree that by the time of the death of Shaykh Ibrāhīm, the portion of the order led by Ibrāhīm's son Shaykh Junayd was not only Shiʿi in character, but militantly so. Since Shaykh Junayd was the grandfather of Ismāʿīl I, Ismāʿīl was head of the Sufi order before he was enthroned as shah, and his conquest of Iran was done through the support of the qizilbāsh supporters of the order, the short answer to your first question is that Shiʿa Islam was so prominent in Safavid Iran because the Safavid monarchs and military class were personally invested in the religion.

But this is only a partial answer, I suppose—again, for a few reasons. First of all, while Ismāʿīl and the qizilbāsh conquered Iran as an already staunchly "Shiʿi" power, the majority of Iranians were Sunni at the time of the conquest. There's a rather pervasive myth (alluded to in u/Khwarezm's question here, which I guess I'm answering as well right now!) that Ismāʿīl converted the entire population of the country at swordpoint. While there's undoubtedly an element of truth to this, it's more likely that the conversion (as judged through the performance of certain defining rituals, like the cursing of the first three historical caliphs) was a gradual process. An anecdote often used to illustrate this line of thinking is that when the Uzbeks conquered Herat in 1587, they granted special privileges and positions of power to the city's Sunni inhabitants—in other words, that more than 75 years after the establishment of the Safavid state, there was still a substantial Sunni population in one of the empire's largest cities! Of course, the conversion process was also not evenly spread out over Iran. The inhabitants of the former Timurid domains centered on Herat, and the western fringe of the empire (populated largely by Kurds and other tribal minority groups), showed much more resistance to the prospect of shiʿitization than did those of Central Iran.

Clearly, if the Safavids aimed to convert the whole of Iran through force, they did a pretty bad job of it. Similarly, the attempt of Tahmāsp's short-lived successor Ismāʿīl II to return the state religion to a form of Sunnism would not have been plausible if the religion were entirely eradicated by the time of his reign. This is not, of course, to say that force was never exerted by the Safavids to pressure communities into confessional change; but that it was only one tactic used over a long period of time. (Equally important, according to Jean Aubin, were moral pressures of the sort used to force the conversion of the Mughal emperor Humāyun during his exile in Iran, as well as financial pressures arising from the lighter tax burdens applied to Shiʿi communities under Safavid rule.)

The other caveat is that while qizilbāsh religion shared many essential elements with Shiʿism, it was decidedly not the Twelver Shiʿism that is Iran's national religion today. The qizilbāsh were heavily criticized by their contemporaries for their religious excesses, which included the (at least quasi-)deification of the heads of the order—Ismāʿīl famously referred to himself as a living god and the reincarnation of ʿAli in his poetry—and esoteric rituals meant to symbolize their devotion to them. (One of the most famous, and certainly among the more bonkers, of these was the charge of ritual cannibalism levied at the Ismāʿīl's soldiers; for more information, see Shahzad Bashir's article on the subject.) The transition to "orthodox" Shiʿism began with the importation of clerics from Lebanon under Ismāʿīl I, and gained a good deal of momentum in the reign of his son Tahmāsp; by the mid-seventeenth century the transition was complete, and the shahs after ʿAbbās I (r. 1571-1629) are often characterized as puppets of the increasingly powerful religious establishment.

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But since, of course, the importation of Twelver clerics into Iran would not have occurred had the Safavid shahs not already followed some form of Shiʿism, I think the short answer still holds here: Shiʿism became the state religion of the Safavid empire not for any reason of state (and certainly not to distinguish Shiʿism as a national Persian religion, as some more nationalist accounts would have it), but because the empire's royalty and military elite had been committed to the religion since before the inception of the state.

BIBLIOGRAPHY

Michel Mazzaoui, The Origins of the Safavids: Šīʿism, Ṣūfism, and the Ġulāt (1972)

Jean Aubin, "La politique religieuse des safavides," in Le schiʿisme imamite (1970), 235-244

———, "Revolution chiite et conservatisme. Les soufis de Lâhejân, 1500-1514 (Études safavides II)," in Moyen orient et océan indien I (1984), 1-40

Rudi Matthee, "Safavid Dynasty" (Encyclopedia Iranica) (2008).

On qizilbāsh ritual and religion, see Jean Calmard, “Les rituals shiites et le pouvoir. L’imposition du shiisme safavide: eulogies et malédictions canoniques,” in Études safavides (1993), 109-150;

——, “Shiʿi Rituals and Power II. The Consolidation of Safavid Shiʿism: Folklore and Popular Religion,” in Safavid Persia (1996), 139-90;

Shahzad Bashir, "Shah Ismaʿil and the Qizilbash: Cannibalism in the Religious History of Early Safavid Iran," History of Religions 45, no. 3 (2006), 234-256.

On the importation of Twelver clerics and transition from qizilbāsh Shiʿism to "orthodox" Twelver Shiʿism, see Rula Abisaab, Converting Persia: Religion and Power in the Safavid Empire (2004).