This is a question that I've been curious about ever since the details of the 9/11 attackers were released. Osama bin Laden was a member of a very wealthy and politically-connected family from Saudi Arabia, and 15 of the 19 hijackers on 9/11 were Saudi nationals, as were and are many other people belonging to or associated with al-Qaeda.
In particular, wealthy Saudi patrons have been implicated in funding and otherwise facilitating the operations of Sunni terrorist groups-including the Islamic State (ISIS/ISIL) most recently. From what I understand, a significant segment of the population of KSA is sympathetic to the ideals and grievances of groups like al-Qaeda and ISIS.
I've read other conflicting accounts of how "modernized" Saudi Arabia is, vs. how much its culture is still extremely rooted in puritanical strains of Sunni Islam (i.e. Wahhabism). There are a lot of wealthy people in the country, but there are also a lot of Saudis-including much of the young adult male population-that face unemployment, poverty, and low social mobility.
Anyway, that was just some of what I had learned already. I would appreciate it if anyone-especially someone more familiar with this country than I-could shed some light on the question of how the Kingdom of Saudi Arabia became such a hotbed of Islamic extremism.
The specific brand of Islam that you are referring to is called 'Wahhabism'. There are many radical versions of Islam but 'Wahhabism', comes directly from and is a conservative and extreme version of Sunni Islam. It is also the version followed by groups such as Daesh (or ISIS/ISIL/IS), Taliban and Al-Qaeda. The followers of this version of Islam don't normally refer to themselves as Wahhabi but the words helps to distinguish them from other brands of the Sunni sect such as Salafism. To understand why this brand of Islam is so closely associated with Saudi Arabia (KSA), you first need to understand that the country is the home of Islam. Prophet Muhammad was born in Mecca and is buried in Medina, both of which are found in KSA. However, the modern country of KSA did not exist until 1932. This is where Wahhabism comes into play.
Wahhabism was started by a man called Muhammad bin Abd al-Wahhab. He was born in the Arabian peninsula during a period when the region's only significance was the religion of Islam. Wahhab through his travels and learning came up with a version of Sunni Islam that was more strict than anything else that existed at the time. During the time when al-Wahhab lived, Islam was in a chaotic state (it still appears to be today because of different sects, ideals, understandings and Qu’ranic translations). So al-Wahhab, with his radical translation, started a movement for reforms. One of the key things that Wahhab pushed for was an end to the polytheistic mannerism that some brands of Islam promoted. Wahhab believed that it was wrong for Muslims to idolize structures representing people who lived, such as tombs, statues and domes (this is why they fight with Shia and Sufi Muslims so much, with an excellent example being ISIS’ activities in Iraq and Syria. This is also the reason why groups like ISIS and the Taliban destroyed ancient statues in Syria and Afghanistan). His movement gained strength when he formed a pact with Muhammad bin Sa’ud, who would soon establish the kingdom of KSA.
This pact is the reason why Wahhabism bears so much weight in KSA today. Working together, bin Sa’ud and al-Wahhab establish the kingdom of KSA. Al-Wahhab benefited from the pact by having the strength to implement his brand of Islam. His ideas had not been popular with many Muslims, because he oversaw the destruction of important Islamic sites such as the tombs of some prophets. He was also condemned for killing Muslims who confessed the shahada, one of the five pillars of Islam. The pact also meant that Wahhabism was the brand of Islam that the kingdom of KSA implemented in its jurisdiction. The pact continued into successive generations of the Sa’ud and Wahhab families, ensuring that Wahhabism remained in the kingdom. This is still seen today in KSA’s radical interpretation of Islam with laws that ban women from banning, a separation of the sexes, a punitive justice system and no tolerance policy on all drugs.
Concisely, the history of the country is the reason why so much radicalism seems to come from it. The government doesn’t officially condone groups like al-Qaeda and ISIS. In fact, it has an adversarial relationship with them. However, rich individuals support these groups financially and its obvious that they (the groups) have significant support in the country.
Focus on Islamic Issues - Cofie D. Malbouisson: This book covers the history of Wahhabism and KSA in a simple way.
Radical Islam and the Revival of Medieval Theology - Daniel Lav covers the issue in a slightly more detailed way.