The rise of militant and extremist Islam was a very gradual process that wasn't really sparked by any single event (though it has picked up in the last century). A lot of times, amongst rather biased scholars mostly, you'll come across the claim that Islam has always been militant and extreme, pointing to the conquests of the Rashidun caliphate and select portions of the Quran dealing with warfare against unbelievers (and ignoring the greater portion of evidence against that interpretation). But what we recognize as militant, politicized Islam today is a largely recent response to colonialism and imperialism from western powers after the fall of the Ottoman Empire, and an Islamic interpretation of western Nationalism. The Mongol invasions were more or less singular events in Islamic history - the Mongols showed up, the Mamluks defeated them, and they were never really heard from again. Even the Timurids, an Islamic dynasty, focused their attention on the East and didn't really disrupt Arabia, the Levant, or North Africa.

But the Muslim Ottomans conquered most of the Arab world, and held it for hundred of years. It was the bridge between the Middle Ages and the Modern World. It was almost like the Roman Empire was to Europe. When it collapsed at the end of World War I, it fragmented the Arab world in many different senses. You ceased to be a citizen of the Baghdadi vilayet of the Ottomans, and became an Iraqi - a citizen of Mandatory Iraq. That led to questions of what the borders of your new country should be. It led to questions of what your country should be - Sunni, or Shia? How should the balance of power between different factions be maintained? What does it mean to be an Iraqi, or a Palestinian, or a Syrian? What do you do with the non-Muslim population? Iraq is a good example of this early reaction, with their outright rejection of High Commissioner Percy Fox's authority. It got to such a point that even King Faisal I, an Arab Muslim, couldn't maintain control even with British help, leading to things like the Simele massacre. Fragmentation was inevitable, and that led to conflicts amongst different Arab powers and political groups that still continues today in many Arab countries. The power vacuum led to jostling for position, revolutions, counter-revolutions, civil wars, and massacres. And lots and lots of propaganda.

The Great Game is another example. Britain and Russia fought quasi-wars for control over Iran and Afghanistan. Russia wanted to keep British power locked up in India, while the British wanted to establish buffer states to protect against a possible Russian invasion through Central Asia. This led to a lot of manipulation in the area, and interference in the government of Persia in particular. They eventually signed a concordance whereby Russia took power over the Northwest of the country, and England the Southeast. The central area around Tehran was maintained as a neutral zone. When the people subsequently rebelled against the Qajar dynasty and attempted to form a constitution and brought in American William Morgan Shuster, Russia invaded and bombarded the Majles - the Iranian parliamentary committee. Things only calmed down when Reza Shah marched on Tehran and established the Pahlavi dynasty. He attempted - and failed - to maintain Iran as an independent nation free of foreign influence. When WWII started, he was close with Nazi Germany (as an ally against Russian and English interference). When he didn't allow England or Russia the use of Iranian railways to send supplies, they conquered the country in about a month and forced him to abdicate. All these events through the Arab world led to a distrust of foreign influence and a strong desire for self-determination.

The one event that it's safe to say significantly influenced Islamist militancy and extremism is the establishment of the oil industry in Saudi Arabia. The Saudis are Wahhabist / Salafist (the names are frequently interchanged, though they're not quite the same), and have been for several hundred years. Wahhabism is a very controversial interpretation of Islam that has strict, more narrow definitions of who is and is not a Muslim - and that narrow definition of Muslim was used as justification by the early Saudis for their attacks on, and conquest of neighboring Muslim powers (such as the Sharifate of Mecca, and the Ottoman empire). After the establishment of the Saudi state, the oil income was used to print and distribute Qurans and Wahhabist literature to mosques around the world, the construction of which was in many cases also funded by Saudi oil money.

All of this comes together to create an unstable region where militancy and extremism means being able to attain and maintain political power, and different groups still fight - mostly along religious fault lines.

There's a bit more to it of course - there's a sense among many Muslims of inferiority in the face of the technological and cultural power of West for example, which leads to fear of a loss of Muslim identity. But by and large it's more mundane, and it's hardly unique. Similar to the extremism that developed in the West before the World Wars with the rise of Nationalism, the Civilizing Mission, the development of democratic and non-democratic governmental systems, and the racism inherent to the justification of colonial governments, it's a product of circumstances. The development of the Islamic State is just as much political as it is religious. To consider extremist Islamism to have roots in a couple invasions in the 1250s is far too reductive and simplistic.

It doesn't make any sense to me. It's rather hard to trace recent Islamism to the thirteenth century. If the fall of Baghdad caused today's Islamism, why didn't it cause that Islamism in 1950, 1920, 1880, etc.? In general, I suggest you look for recent causes to explain recent effects.

For more information about how the Mongolian conquest affect Islamic thinking, I recommend the book "heaven on earth: a journey through shari'a law from the deserts of ancient arabia to the streets of the modern muslim world" by Sadakat Kadri. It has a good chapter on the fall of Baghdad and its effect on Islamic jurisprudence.