This post turned out to be much longer than I anticipated, so I split it up in two parts. Sources noted at the end of the second post.

The context

Before providing a tentative answer to this question there are some important things I'd like to point out.

1. Firstly, the Iranian Revolution occurred relatively recently. There is a considerable portion of the Iranian population that was forced to go into exile and whose immediate family are still living daily through the effects of the revolution. As a result, discussing the Iranian Revolution, even in an academic setting, can degenerate quickly into very heated argumentation.
2. Secondly, much of how one approaches the causes and the success of the Iranian Revolution depends on a certain methodological framework. There have been a number of different attitudes to answering these questions: were the main causes economical, political, religious in nature? In my answer, I'm going to try and show how there were numerous problems with the late Pahlavi regime, which culminated in a broad base of support for the deposing of the Shah.
3. Thirdly, due to the two points mentioned above, there are going to be people who will disagree (vehemently) with my perspective. That's alright. My Persian is not what it used to be, and I rely mostly on the literature I had to read for a class I did back in undergrad. I'm going to focus specifically on the role of some specific socio-political actors during and before the Revolution: they include the merchant middle class known as the bazārīs, as well as other kinds of anti-monarchical opposition and finally, the role of the Shi'a clergy.

Now that this is out of the way, let's get to the answer. I believe that there are number of specific factors you can look at to understand why the Pahlavi regime failed. First of all, I want to talk a bit about the so-called bazārīs.

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The bāzārīs

The political influence of the people in charge of and working at the markets (Persian bāzār) goes all the way back to the conquest of Iran by the Mongols and the subsequent founding of the Ilkhanate empire (1263-1336). As you can imagine, this is a very long time ago, but it is already at this period that the rulers of Iran, whose governments were generally decentralized, depended strongly on the assistance of the merchant class. These power of these merchants increased during the following Safavid and Qajarite periods: as an example of how influential the bazārīs could be, they played a pivotal role in the success of the Tobacco Protest of 1896, which drastically shifted the balance of political power in Iran. One of the consequences of the Tobacco Revolt was direct Russian interference in Iran, which later brought Reza Shah, who was a colonel in the Tsarist army, to power in 1921.

Anyway, the bazārīs played a similar role in the years leading up to the Revolution. As an apt demonstration of how interconnected global events are, a major reason as to why the Revolution happened has to do with the Arab-Israeli War of 1973 and the subsequent oil boycott by the OPEC countries.

Virtually overnight, Iran('s government) profited immensely from the dramatic increase in oil price. The Shah spent the vast majority of this newly found income on military expenditures, which increased inflation, income inequality, reliance on foreign experts as well as a direct demand for a number of limited resources (particularly cement, decreasing the amount available for housing projects, significantly).

The wealth gained from Iran's economical reliance on oil exports benefited a relatively small group, mostly consisting of the Shah's inner circle and close associates, who lived a lavish, Western lifestyle. However: inflation, overcrowding and the Iran's increasing economic reliance on a single product particularly hurt the aforementioned bazaris, who came to play an important in bringing down the Shah's rule.

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Anti-monarchical opposition

An important reason as to why the Iranian Revolution was able to find widespread support was due to the fact that it began as a wide, anti-Shah movement. This movement included parts of the religious establishment, but also socialists, communists, as well as what you could call "bourgeoisie" intellectuals. The members of this movement disagreed amongst each other what the goals of the Revolution should be, beyond the deposal of the Shah.

In the years preceding the Iranian Revolution, not all opposition was religious in nature. Many different people of various backgrounds and ideological convictions were against the Shah. Some of them advocated the replacement of regime with an Islamic one, but this was not necessarily the only one. Others advocated the implementation of some aspects of Islamic jurisprudence, but in a modernized setting. There are two important writers, Ali Shariati and Mehdi Bazargan, who combined their Islamic theology with modern political philosophies: Shariati mostly looked towards Marxism as a sources of inspiration, whereas Bazargan was a liberal, more concerned with justice as a whole. Both of them spoke to large segments of the Iranian population.

Apart from these two authors (and there are more, but they are certainly two of the more influential) there were also other, non-Islamist movements, in the sense that they advocated the replacement the Shah's regime with a secular one. Such movements include the communist Tudeh Party, founded in 1941, which had played an important role in the Mossadegh government of the early 1950s.

Although their influence had peaked during the 1960s and had soon began to wane, the Tudeh party helped organize mass demonstrations and strikes in the final months before the Revolution. Perhaps more importantly, there were several off-shoots of the Tudeh party which had become radicalized and drew support from other population groups. In short, opposition against the Shah was characterized by a broad base, which did not necessarily all share the same goals.

/u/Kiviimar's answer is very good and comprehensive, and stresses the factors that I think are most important in answering the question at hand: that the Iranian revolution was broadly based, that opposition to the Pahlavi regime centered not only on cultural issues (like the importation of Western culture) but also on socio-economic issues and economic mismanagement, and that figures like Khomeini were able to position themselves as defenders of the people against this coalition of popular grievances. Eventually, through adept political maneuvering, he was able to come out on top of his factional rivals.

I would add two observations to this: the first is regarding the role of religion, and the second the nature of the Shah's regime itself.

First, that religion (and especially Shia religion) was the primary basis around which the average Iranian organized their lives, especially outside of the cities. Until the Shah's reforms of the 1960s, the clergy was the main mechanism for day-to-day social and economic disputes in what might be considered the realm of "municipal court" in other systems, as well as control over educational facilities and monetary religious endowments. They were therefore not so much "quietest" as they were willing to work with the government hand-in-hand, a kind of "you handle your domain and I'll handle mine" sort of arrangement. One of the many ways in which Reza Khan and his son, Mohammad Reza Pahlavi, destabilized the political system and offended the sensibilities of the average person was to openly challenge and destroy the Imam's base of power in society, primarily by supplanting them in the legal system.

At this time, drawing on a history of social dissent going back to the Tobacco Riots and the Constitutional Revolution, many clergy switched from "going along to get along", or simply decrying Western influence without making it explicitly about the government, to attacking the Shah openly. When Khomeini was exiled for the first time, it was for such a personal attack on the Shah. I say this to emphasize that there was a political divide between the government and the clergy, as well as a religious one, so when the clergy got involved in politics, it was easy for them to draw support from the average person, who had their own grievances against the Shah and his westernized government. This is one reason that the ultimate government form ended up being an Islamic government.

Second, it is true that the Shah attempted to modernize the country and succeeded on many levels, but it was often these same reforms that caused opposition against him. For example, his attempts to create mechanized farming had the result of consolidating land into the hands of large factories, and forcing people who previously lived in villages as subsistence farmers to join the ranks of the masses of out-of-work urban poor. The economic strife of the late 1970s exacerbated this significantly. This helped create a further divide between the people and the government, one which was often expressed in religious terms, as the Imams saw these people as their natural constituents and began to advocate on their behalf.

Ultimately, while the Shah modernized the economic system with his "White Revolution" (to stave off a Red Revolution, he said), he never modernized the political system. One of Ervand Abrahamian's major arguments in Iran Between Two Revolutions is that the Shah's reforms helped create a series of modern classes - educated university students, a modern middle class, a dislocated and disempowered religious establishment, a mass of poor urbanites - that had every reason to oppose him. In order to clamp down on dissent, he did not liberalize the political system, and maintained the use of secret police organizations like SAVAK to violently repress anyone - usually students - guilty of subversive thought or possessing revolutionary/communist literature. This created only more resentment.

Finally, I would like to address something I think the other answer misses, namely: if an Islamic republic was an unexpected outcome, what DID people expect? The answer is that the most popular political currents in the country at that time were various forms of nationalist and, perhaps moreso by this time, leftist thought. In other words, what was expected was some form of democratic or socialist government, and certainly one which was primarily aimed at redressing the social grievances, the problem of poverty, and economic development. I've written elsewhere on the importance and the history of the Iranian Left in the history of the revolution. Whether expressed as Marxism, or more commonly as Shari'atis fusion of Marxist and Shi'a Islamic ideas, most educated Iranians in the 1970s who were politically active were animated by some form of nationalist or socialist politics. Islamism was a relatively small part of that movement, which consisted of "God-Worshipping Socialists" like Shari'ati and conservative clerics who railed against the regime using the language of revolution and nationalism. In short, some combination of Marxism and revolutionary Shi'a socialism was the most popular form of political expression.

Gonna quote myself from the previous answer here, but highly relevant: "So popular was the radical politics and rhetoric of the Left that the Shah's regime engaged in a similar project of plagiarism and co-opting popular leftist ideas to put them in the service of dictatorship. In the 1960s, the Shah began to engage in what Matin-Asgari calls “authenticity politics,” in response to popular discourse that sought to defend Iranian culture against the “technological, economic and cultural domination of 'the West.'” The state became more and more involved in the production of an intellectual culture explicitly cultivated to counter the influence of the opposition, which included the creation of organizations such as the High Council of Culture and the Arts that employed former Marxists, and especially radical Maoists, in key positions. This language only increased as the Shah warmed up to China in the 1970s, and some Iranian Maoist groups repeated the propagandists' claims that the Shah was “an anti-imperialist leader defiant of both superpowers,” to the detriment of their credibility with the opposition."

(This also accounts only for those who were politically active, the middle classes and educated people. People living in villages or outside of major cities did not even know the revolution was happening until the Shah himself foolishly proclaimed that he had "heard the voice of the revolution" on the radio in the middle of all the unrest, thus admitting to the entire country that a revolution was happening and redoubling opposition against him)

It's worth noting that in the years leading up to the Revolution, 1971-1979, opposition against the Shah had been almost completely crushed and driven into exile outside the country. Opposition flared up again at the hands of guerilla organizations like the Mojahedin-e Khalq and the Fedaiyan, which all espoused a combination of Marxist and Islamist ideologies. Beginning with the Siakhal Incident in 1971, where guerillas killed three policemen and freed two previously arrested militants, a string of increasingly brazen attacks on government property and the lives of officials helped pushed the Shah's regime to the brink.

The first stirrings of the revolution came in 1977, after the Shah - who seemed to believe that he had crushed all true dissent to him and won over the hearts of his people - relaxed political restrictions and the more strict police controls. This iwas in part because of the campaign being run against him in the international media by exiled Iranian students, and U.S. President Jimmy Carter was leaning on him to do something to counter this image. However, ultimately, the shah granted these concessions because, to quite Abrahamian, "he was confident he could weather the storm...he had deluded himself into thinking that he enjoyed overwhelming public support. He boasted privately...that the only people who opposed him were the 'nihilists'". He would prove to be quite wrong.