I would suggest reading The Looming Tower: Al-Qaeda and the Road to 9/11 by Lawrence Wright. It describes the events that led up to the 9/11 terrorist attacks, including the genesis of anti-Western sentiment and modern Islamism in Egypt in the 1950's and 60's. Importantly, Wright spends a good deal of time detailing the life of Sayyid Qutb.

Qutb came to the United States from Egypt in the 1940's to study. During his travels across the country, he was struck by what he thought was the overt sexuality of American women, and the primitive nature of American culture/art.

Upon returning to Egypt, Qutb joined the Muslim Brotherhood, in part as a rejection of Western values based on his experiences in America. Shortly thereafter, the pro-British monarchist government in Egypt was overthrown in a coup led by Gamal Abdel Nasser. The Brotherhood welcomed the coup as an opportunity to establish an Islamic government. It soon became clear that Nasser's government was in fact a secular nationalist government and not Islamic.

Nasser offered Qutb a role in the new government, but as it was not compatible with Qutb's Islamic ideology, he refused. Soon thereafter, in 1954, the Muslim Brotherhood planned to assasinate Nasser. The plot was foiled and Qutb, along with other members of the Brotherhood, were jailed. It was during this imprisonment that Qutb cemented his anti-western and anti-secular claims. Today this philosophy is known as Qutbism.

Qutb was briefly released from prison, but was soon arrested again and sentenced to death for plotting to overthrow the government. Qutb was executed (read: martyred) in 1966. But not before he laid the groundwork for modern Islamism.

I'd just like to add "Islam" itself is not anti-western. Instead we find anti-western sentiments crop up in certain modern ideologies, but the religion itself is not anti-western.

Part of this was a result of European colonialism. In fact there was a lot of jihad revivalism centered around expelling European colonizers, like what was seen in parts of West Africa. As well there was a rise of people claiming to be the Mahdi signaling the end of times, like Muhammad Ahmad (the Mahdi) in Sudan, Mirza Ghulam Ahmad in India, and The Bab in Persia. Many parts of the Muslim still have the fresh scars and memories of colonialism. I did I was reading a Karen Armstrong interview the other day and she talked about how many places reject modernity because modernity was thrown on them by European colonizers, so there is no sense of connection or authenticity about it. So much of the rejection of the West in certain Islamic ideologies is a legacy of European colonialism of Muslim lands.

I think the anti-Western philosophy which you are referring to largely developed in reaction to European imperialism. Egypt became a cradle of Islamic extremism, producing philosophers such as Sayyid Qutb which others have mentioned. This is hardly surprising considering how the British Empire dominated Egypt's political and economic affairs throughout the 19th and much of the 20th century. Whole cities were built outside of Cairo to accommodate foreign workers and investors who flooded into Egypt to take advantage of lucrative economic opportunities. Under imperial pressure, the Egyptian ruling class (the Khedives) were forced to grant concessions such as exempting westerners from paying any taxes, except those levied on property, and immunity from trial in Egyptian courts. Furthermore, the Khedives, at the behest of the foreign elite, raised exorbitant taxes on an increasingly impoverished local population. It is only natural that in such an atmosphere a strong anti-Western sentiment developed.

Thinkers such as Jamal ad-Din al-Afghani in the 19th century looked to Islam as a means of uniting the Muslim world against imperialist aggression. He argued that Islam could be used a sociopolitical ideology, rather than just a religion. This became known as pan-Islamism, and it was later adopted by a religious scholar named Hassan al-Banna, who went on to found the Muslim Brotherhood. As the European powers pulled out of the Middle-East, the bloody legacy of colonialism lingered on through the numerous violent dictators which quickly seized power in the region. It was during this period when anti-Western philosophy began to mutate into what we might recognise today as Islamic extremism. The Muslim Brotherhood was banned under Colonel Nasser's regime in Egypt, leading many of its members to turn to violence. Sayyid Qutb, for example, joined the Muslim Brotherhood in 1950 and was subsequently arrested, imprisoned and tortured. During his time in prison he wrote his revolutionary manifesto 'Milestones' which has had a huge influence on extremist thought. Many others who held similar beliefs fled to Saudi Arabia, the ideological homeland of a puritan branch of Islam known as Wahhabism, where they found eager financial backers.

This is a very general overview of the pertinent events. To learn more about Islamic extremism and how it emerged in response to Western imperialism, it's worth looking into the Indian Rebellion of 1857 and the history of Iran. I would thoroughly recommend 'No God But God' by Reza Azlan, which is an excellent, if somewhat brief, history of Islam.