Well, its tricky to say. To what extent is anything "caused" by the Cold War? The Cold War is just the term we use for the political system post WWII until the end of the Soviet Union. The relationships that every country had with the dominant superpowers was affected by the adversarial relationship that they had, and this was true of the Middle East as well. Ill try to give a rundown on the topic, but remember this is a very general overview of a complex place, time, and theology. Ill be happy to clarify anything!

Another comment mentioned Sayyed Qutb, and I think he's probably the best man to start with when discussing the beginnings of the modern "fundamentalist" movement, however I take issue with several of his points, especially regarding Qutbs ideas and "radicalization".

First, Qutb, while influential, was by no means creating his ideas in a vacuum. He drew heavily on several previous Islamic thinkers, as will those who were in turn influenced by his ideas, such as Bin Laden; scholars such as Ibn Taymmiah, Jamal Al-Din Al-Afghani, and Muhammad Abduh. It should be noted that these men would not be necessarily considered "fundamentalists". Taymmiah was writing mainly about the Mongol conquests and jihad in the sense of resisting the Mongols, and would contribute the early beginnings of "Takfiri" doctrine. Abduh and Afghani were progressive Muslim scholars of the Nahda (Arabic Renaissance) who believed that Islam and Islamic thought must adapt to keep up with the science of the western world; they would argue that a Muslim must not blindly accept the legalistic schools of Islam and instead use their own minds to read and interpret the Quran (within reason). Qutb will also, later in life become a prominent member of the Muslim Brotherhood, an organization which has existed since 1928 and had created its own sets of ideas about an "Islamic" state. The Muslim Brotherhood was frequently cracked down upon by the Egyptian state, both under the Farouk monarchy period as well as during the Nasser period after Gamal Abd El-Nasser's coup.

Qutb, as a prominent educator, was granted leave to study abroad in America and would recieve a masters from the University of Greely, Colorado. Upon his return from the west, he would write "The America I Have Seen", in which he describes America as a culturally decadent place. In it he is critical of many American values and pastimes, such as the comparatively loose women, individualism, racism, violent sports, etc. However, we need to keep in mind that, while the book is "Anti-American" in the sense that he does not like the culture, he is more writing as a warning to Egyptian audiences that this is not what we want our culture to become. After he returns to Egypt, he would join the brotherhood, so once again, these ideas he has developed happen in absence of a cold war context and come from a more cultural fear that Arab society will be polluted by western values.

Nasser's coup takes place in 1952, overthrowing the pro-West Farouq government. However, until 1967 Egypt will remain more or less unaligned in the Cold War, with Nasser skillfully playing off the superpowers against each other and accepting funds from both. At the same time, Nasser will enforce a brutal crackdown on the Muslim Brotherhood who, as a secular leader and Arab Nationalist, he sees as a major threat. Qutb (though first offered a job by Nasser in an attempt to co-opt him towards the regime) was a part of a plot to assassinate Nasser, is imprisoned and brutally tortured. The assertion that the Egyptians were trained by the CIA at the time is ludicrous, as the Egyptian government was becoming more and more pro-Soviet at the time, especially following the Suez Crisis.

During this period, Qutb would write his two most important works, "In the Shade of the Quran" and "Milestones". Here he details his idea of an "Islamic State" ruled by sharia law. He is equally critical of Western democracy and Eastern communism and Arab secular nationalism, believing that the only way in which a society could ever be justly governed is with the principles of Islam. He also argues that all of the worlds regimes, Arab especially, are living in jahhaliya (Ignorance). He argues that the only way to free the world from jahhal is to aggressively, violently if necessary, spread Islam and Islamic values to the state level and form a true Islam. He would be released by Nasser in 1964, but arrested shortly afterward, tried, and executed.

So here with Qutb, a man who is generally credited with influencing and creating the ideas of violent, aggressive jihad, there is very little "Cold War" style involvement. Qutb is mainly commenting on the Arab body politic and the Arab state and Arab Nationalism. He does not like the West (or the East), but he does not concern himself with the Cold War per se.

However, the Cold War politics of the region will have a major influence on the development of jihadist groups over the next 40 years. Following the 1973 war, the Arab region as a whole will be fairly firmly in the US camp in the cold war. With the exception of some small communist movements in several countries such as Lebanon, and states such as South Yemen and Syria, communism will never gain a real foothold in the region. Especially following the peace treaties between Israel and Jordan and Egypt, the regimes of the region will receive massive amounts of US aid to shore up their authoritarian governments. They will all, as Arab secular states, continue to crack down upon Islamists of any form. However, after the Iranian Revolution, which quickly became a theocratic Islamist revolution, the Islamists of the region were inspired to rise up against their US backed leaders. The 1970s and early 80s would see a good deal of state vs Islamist unrest across the region. The Saudi state would also crack down upon Islamists in their own country, as these Islamists were generally affiliated with the generally more moderate Muslim Brotherhood and not the Wahhabism of the state.

The Soviet Union would invade Afghanistan to prop up the communist Afghanistan government in 1979. The United States would flood money into Afghanistan to Afghani mujahideen to fight Soviet troops. Now this is a topic which deserves its whole own answer, but as I said at the beginning this is a very general overview. The US would partner with the Pakistani ISI and the Saudi secret services led by Prince Turki Bin Faisal to run weapons and funds into Afghanistan. All of these intelligence forces would be working at somewhat cross purposes to each other with the same goal. The Muslim world was extremely galvanized by the Afghan War; many saw it as an existential crisis, with godless communist forces fighting a brave Islamic resistance. Many around the world would answer the call of Jihad in defense of Islam, and many of those who remained home would send funds independent of any state government.

Osama Bin Laden would be one of these. Bin Laden was extremely wealthy, and would form a group called the Arab Afghans to take part in the Afghan war. Bin Laden was largely self funded, however, he may have received some funds or assistance from Saudi officials due to his Saudi nationality and connections with the Saudi state. The Arab Afghans were not particularly effective in the war itself (in fact, many Afghan Mujhideen found them to be a nuisance overly concerned with becoming martyrs instead of fighting, and they were generally ignored except when funds were needed). However, it really needs to be emphasized that to many around the Middle East region these men were heroes, and those who went began to display a religious fervor commensurate with their growing reputation. After the war ended and the Soviets retreated in 1989, Bin Laden (and his partner Ayman Al-Zawahiri, an Egyptian also heavily influenced by Qutb's work) were forced to decide what to do with the network they had built and those Arab mujahideen who wished to stay and fight. Many of the fighters, Zawahiri included, wished to go home and take a violent fight against their secular "godless" regimes at home such as Egypt. Bin Laden however would argue that to succeed in overthrowing the Arab States living in jahhal, the mujahideen must alter their goal. He would argue that they should not focus on the "Near Satan" of the Arab states, but instead focus on the "Great Satan" of the United States, which was the true power and backer of the Arab regimes. Due largely to the fact that Bin Laden was the primary financier, many of the other Mujahid would agree and form the beginnings of Al-Qaeda.

At this point we are getting fairly into 20 year territory, but I'll just leave it that Al-Qaeda (and the effects of the later US war against it) is the template organization and theology for much of the (violent) Islamic fundamentalism that we see today. I have brushed over a lot of more existential questions here, not the least of which is "What is Fundamentalism?" (do we count the vast majority of non-violent Salafis?). Ive also brushed over so much important history, but like I said, I can elaborate on specifics if asked. If you're looking for some reading on this, some recommendations:

"The Looming Tower" by Lawrence Wright - an excellent history of Al-Qaeda, beginning with Qutb and ending with the aftermath of 9-11

"Radical Islamic Fundamentalism: The Ideological and Political Discourse of Sayyd Qutb" by Ahmed Moussali - The go-to end-all-be all-book of the life and theological thought of Qutb.

"Ghost Wars" by Steve Coll - An excellent history of the CIA involvement in Afghanistan

Edits: Because I don't spell or grammar no good.