Answer:
Juhayman Al-Utaybi was NOT an extremist. And technically, he wasn't even a terrorist. Saying he was simplifies the answer - but also greatly falsifies history.
The situation of the Middle East and especially in the Kingdom of Saudi Arabia in the late 1970s was extremely complex. Westerners love to make Middle Eastern history easy and to divide it into black and white, good and bad, right and wrong.
To really understand what happened and get a grasp WHY the "Siege of Mecca" happened, you have to get an understanding of how things were back then in the strange place that is the Kingdom of Saudi Arabia.
Explanation:
Al-Utaybi was a bedouin, as were most of the approximately 450 men, women and children (!) who accompanied him into the Grand Mosque on November 20, 1979. Bedouins are the nomadic and tribally organized natives of the Arabian peninsula, while the Al-Saud family - who had founded the country already in the 18th century - are part of the "urban" populace that dwelled in fortified cities.
Saudi Arabia was unified and founded (in 1928 the former, in 1932 the latter) by Abdulaziz Al-Saud, who later became the first king, with the help of an army of fierce bedouin warriors - the so-called Ikhwan or "Brothers" (no relation to the Muslim Brotherhood). He needed them to unify the huge territory, and he promised them to make them participate in the new country that was about to be born.
Yet, when he was finished with his task and began to sign international treaties with the superpower of the time - the British Empire - his warriors rebelled. They couldn't understand how the new King could make a pact with people who for them were non-believers. They rebelled - and were crushed in a memorable battle. Juhayman's father and grandfather participated in that battle, and the young boy grew up - as did many other bedouins - with that memory.
So, one motive was revenge and justice toward the bedouins. But there were others. One of the most important was that in Juhayman's eyes, the Saudi regime had lost their legitimacy to rule the country. He thought that the oil boom of the 1960s and 1970s and the stark increase of petrodollars after the oil crisis of 1973 (that made the price of crude go up almost tenfold) brought too much Western money, Western goods - and Western ideas into the country. And the ones who had profited most from this were - the members or the Royal family. Their spending sprees in Europe were as legendary in the 1970s as they are today (google that, you'll be surprised how little has changed).
In other words: for Juhayman, Saudi Arabia was developing way too fast and to the wrong direction. Occupying the Grand Mosque, he demanded that Saudi Arabia returned to the religious and social values of ancient Islamic tradition. Saudi Arabia is not only home to the two most important shrines of Islam, Mecca and Medina: the Kingdom is the physical heart of Islam (at least to the Sunni part of it, while Shiites revere other sites, most of them located in Iran). That is mainly what Juhayman and his people wanted to achieve.
Juhayman was not a terrorist, he was more a rebel. This is because he did not threaten or kill anyone - except for the Saudi Military when they began to attack the Grand Mosque after a few days. On November 20, when he and his people entered and occupied the mosque, there were around 100.000 pilgrims inside (let this number sink in, please). Why so many? It was the end of the Hajj season (the sacred journey to Mecca that is one of the pillars of Islam and a duty of every Muslim) and Mecca was still filled with people from all over the world. However, after less than 12 hours, all the hostages were released, and only those who had started to believe in Juhayman's cause remained.
The 14 following days of what became to be known as the "Siege of Mecca" should prove a disaster for the Saudi regime on multiple levels: military, intelligence, political and even religious (the latter two are very close in Saudi Arabia still today). As a reaction to this attack on the Holiest shrine of Islam, the Saudi king Khaled and his Crown Prince Fahd had two options: understand what the rebels wanted, practice a tiny bit of self-critique and try to quell the differences in the country. Instead, they chose to execute the rebels and become more religious than what Juhayman had ever wanted. Saudi Arabia became the arch-conservative country that it mostly is in still today - even if things are changing rapidly right now, at least on SOME levels...
There is an excellent recent documentary about the Mecca Siege that you can find at mecca1979 dot com - the filmmakers researched for 6 years to do it. It was also shown several times on PBS, albeit a shorter version.