hi! there's lots of room for more discussion on this, so not discouraging further input here, but fyi, there have been some lengthy posts in the past that may be of interest:
What is the history of (modern) Islamic extremism? When did this all start and why? (x-post from /r/history) - featuring u/yodatsracist
Was radical Islam an issue prior to the 20th century? If not, what was the catalyst that made modern Islam have a large radical following? - featuring u/Dark-Ulfberht
Is there a historical reason for the rise of Islamic extremism as opposed to, say, Christian or Jewish extremism? - featuring u/agoyalwm and u/yodatsracist
Why did Islamist Terrorism not really become that big of what it is today until after the Cold War? - featuring u/crassy, u/CptBuck, and u/mystical-me
Is modern Islamic fundamental terrorism a product of the Cold War and decolonization, or are there notable examples of such terror attacks prior to World War 2? - featuring u/CptBuck and u/tropical_chancer
What actions sparked the modern form of terrorism, specifically the terrorism of Muslims killing Muslims. - featuring u/Smilin_Dave
What were the main historical forces that led to Saudi Arabia being-arguably-the number 1 source of (Sunni) Islamic extremism and terrorism? - featuring u/Joplings_Knuckles
if you have follow-up questions on any locked post, just ask them here & page the relevant user by including their username
A good political figure to start with would be Sayyid Qutb, his time, interaction in America and his writings after and with The Muslim Brotherhood. His writing formed a lot of the ideological basis for modern Islamic extremism. Osama bin Laden and his mentors for example were deeply influenced by Qutb's writing.
The Political Thought of Sayyid Qutb The theory of jahiliyyah by Sayed Khatab
You can probably get the above through an ILL with your local library.
Social Justice in Islam Sayyid Qutb
He is genuinely one of those few figures in history had he not been born or died at an early age, the world would likely be a significantly different place.
The main ideological starting point is Muhammad Ibn Abd al-Wahhab in the 1700s. From the perspective of Islamic theology and jurisprudence, he deviated majorly from the existing Islamic tradition up until that point.
Without going too into detail about the madhab system, Sunni Islamic law had pretty much been canonized into four schools of thought by the 900s. With minor alterations due to changing contexts, the guidelines of these schools were the bedrock of what Muslims considered normative Islamic behavior, at both the governmental and personal levels.
Well when Ibn Abd al-Wahhab comes along in the 1700s, he determines that the reason the Muslim world was declining politically and socially at his time was that Muslims had deviated from the "original" truth of Islam. He thus attempted to bypass the Islamic legal tradition and extract rulings on his own based on his own understanding of what the Quran says and what the Prophet did. A good analogy to what he did is the US Supreme Court making rulings without looking at any precedent at all and basing rulings only on a literal reading of the Constitution and a few other documents from the Founding Fathers.
Long story short, Muhammad Ibn Abd al-Wahhab opened up the door for people to begin to interpret Islam independent of the existing legal tradition and oftentimes without the rigorous training it takes to become a qualified jurist (he himself was not particularly well trained in traditional Islamic thought). If laymen can interpret the Quran however they want, some will interpret it as a book of war, giving them a blank check to kill, terrorize, etc even if it contradicts traditional Islamic thought.
Further reading:
The Canonization of Islamic Law by Ahmed El Shamsy
Islamic Legal Theory by Wael Hallaq