I shall write this answer in 2 parts; the background to Isma'il's rise and the details of his rise themselves.

Part One

The religious life of the eastern Islamic world in the years since the Mongol conquest underwent a remarkable reorientation from previous practice. The destruction of the 'Abbasid Caliphate helped to accelerate trends which had begun in the early 13th century: a move away from textual, legalistic norms and the rise of both a worldly type of Shi'ism and a proliferation of heterodox movements seeking to radically reinterpret (or in some cases supersede) Islamic teaching. Patronage of important Sufi orders and Sufi shaykhs became an important tool for rulers to legitimise themselves; there was a certain spiritual prestige in gaining the du'a-ye khayr (prayer or blessing) of noted holy men. Intermarriage between Sufi and royal families was common, and Sufi orders became immensely influential and poweful.

Concurrently with this was a rapid rise in the veneration of Ali and Ali's family. Ali was increasingly being cast in the mould of a heroic saint by Shi'is and Sunnis alike; patronage of Shi'i and Alid shrines and sites was performed by both Shi'i and Sunni dynasties- such as the Timurids- and a low culture centred around Alid narratives began to grow up. Storytellers would recite works such as the Abu Muslimname (in which Abu Muslim was, somewhat bizarrely, recast as a fighter for the Alid cause), Junaydname, Mukhtarname, Aliname and the Khawarname. As can be seen in Ali-focused stories such as the latter two, Ali himself was recast in superhuman terms as the greatest warrior of the early Islamic cause. Chivalric (futuvvat) circles became increasingly centred on devotion to Ali; the most comprehensive description we have of the initiation rituals, from Va'iz-e Kashifi at the end of the 15th century, describes entry to such circles as centred on knowledge and devotion to the Prophet's family. Such a culture almost naturally led to a great deal of high esteem given to sayyids, the descendants of Muhammad via Ali and Fatima (although descendants of Ali by other women, such as the Hanafiyya bloodline, were also venerated to a lesser extent), to the point that a great many Sufi lineages in the 14th and 15th centuries claimed a sayyid lineage to enhance their charismatic order.

This type of religious culture often expressed itself in far less respectable and orthodox ways, however. The examples of the Mar'ashi and Musha'sha orders shows that these venerated Sufi lineages would often seize political power for themselves; the former took control of a large chunk of Tabarestan in 1358, while, while the latter seized control of Basra in the late 15th century. Other heterodox, often millenarian (due to the proximity to the Islamic millenium in 1591/2) movements flourished: the cabbalistic Hurufis, who sought a greater spiritual meaning in the Qu'ran through a gnostic analysis of the letters of the Arabic alphabet; the Nuqtavis, a Persianate movement seeking to supersede Islam through a return to a Mazdean-Islamic blend in a cyclical conception of history, wherein Arab and Iranian domination would alternate; and the Safavids themselves, to whom we shall return later. These movements all have a common culture of messianic and millenarian expectation: the Musha'sha shaykh Mawla 'Ali claimed to be the reincarnation of Ali, for instance. Most commonly, these movements would claim to be the herald or the incarnation of the Mahdi, the expected redeemer of Islam prior to the apocalypse. We thus see the transformation of many types of Sufism from an ascetic movement to a militant, politicised phenomena, drawing upon their immense charisma and popular support to effect religious and political change, alongside more heterodox groups, with a common belief in the imminence of the apocalypse.

The emphasis on sayyid and saintly veneration led to a culture of shrine-based religiosity. Huge amounts of material patronage and tax breaks were given to a wide variety of shrines of Sufi shaykhs and saintly tombs- the latter normally claiming to house an Alid hero or family member. And one of these shrines was the Safavid shrine of Ardabil, which became increasingly important throughout the 15th century. Founded in 1301 by Safi al-Din, who took over the Zahidiyya lineage upon the death of its shaykh Zahed Gilani, the Safavid order was merely one of many prosperous urban Sufi orders in the 14th and early 15th centuries. Its shrine in Ardabil gradually became the centre of an economic network of trade and patronage across Azerbaijan and eastern Anatolia, and became one of the more prominent Sufi orders in Iran.

This began to change with Shaykh Junayd. In 1447, upon the death of Shaykh Ibrahim, the order split between its Ardabili and Anatolian branches. While Shaykh Ja'far in Ardabil continued as the head of an ordinary Sufi order until the 1470s, Junayd in Anatolia began to transform his branch into a militant movement, converting many Turkmen tribes in eastern Anatolia to his cause. Leading them as a ghazi (holy warrior) on raids against the Christian powers of Georgia and Trebizond, he cultivated a reputation as a heroic religious warrior. His son Haydar made his followers wear a red, twelve-gored taj (crown) symbolising the Twelve Imams, and reconquered Ardabil in the 1470s. Here, we see the transformation of the Safavid order into a far more militant, heterodox Alid movement; Junayd is reported to have been venerated by his followers as God, with Haydar acclaimed as the "son of God"; at the very least, the Safavid shaykhs were certainly venerated as the Mahdi. Haydar bolstered his credentials by intermarrying with the Aq Qoyunulu royal house (emblematising the royal-Sufi links so common to the period), but he eventually fell foul of them when they grew too suspicious of his increasing power. A series of conflicts conducted by him and his son Soltan Ali against the Aq Qoyunulu and the Shirvan-shahs led to their deaths and the forcible removal of the Safavids from Ardabil. Haydar's youngest son Isma'il was spirited away as a child by the Safavid devotees- the Qizilbash- and sheltered, hidden in Lahijan, protected by its Zaydi ruler.

I give this long section of background because this religious context is the background against which Isma'il's conquest occurred. It was a time of heterodoxy and a time of "ghulat movements; "exaggerated" forms of Islam. The religious life was centred around shaykhs, shrines and saints, with holy men and holy sites being venerated within a historical memory largely centred around Ali and Ali's family. The Qizilbash themselves represented such a movement, and what we know of their beliefs reflects this. Doctrines such as tanasukh or metempsychosis (a belief in the transmigration of the soul), incarnated divinity, and the belief in the creation of an Alid utopia on earth were important parts of many of their beliefs. This was the kind of movement that Isma'il's conquests represented- which is why his adoption of legalistic Twelver Shi'ism often seems so strange to us.