I think the characterization of Islam as "turning inwards" is an interesting one that has to be understood in context.

In one sense, the empires of Islamdom after the Mongols were, if anything even more vigorous in their outward conquests. The Ottomans, the Safavids and the Mughals, the three main "gunpowder" empires of Islam in the centuries after the Mongols, all conquered huge swathes of territory, particularly the Ottomans in the Balkans and towards central Europe and the Mughals into India.

(As an aside, Ottoman Europe I think is misunderstood. Because the siege of Vienna was so much later I think there's a popular thinking or assumption that their European territories were somehow the product of a great buildup that came after consolidating the Middle East. Actually it was the other way around, the Balkans and Greece were integral Ottoman territories almost a century before they would conquer the bulk of their Arab territories.)

Aside from military conquest these were also vigorous and prominent international powers who were deeply involved in international trade and diplomacy.

So Islamdom didn't really have any inward turn to speak of.

Intellectual Islam, however, as a religion and incubator of philosophical thought, has been characterized after the Mongol conquests as having an inward turn after the Mongols.

To a certain extent some of this thinking predates the Mongol conquests. One of the most prominent thinkers of this inward turn or trend was Al-Ghazali, who produced what is today regarded as the work of Islamic anti-rationalism par excellence The Incoherence of the Philosophers in the 11th century.

Perhaps ironically, the other roots of this can be found in the so-called "Sunni Revival" of the 11th and 12th centuries. This is a large topic on which there is much debate as to whether or not it even occurred (comparable to the European renaissance in some senses) but suffice it to say that as an intellectual movement these centuries produced a largely homogeneous orthodox consensus within Sunni Islam and amongst Sunnis.

So why are the Mongol Invasions important if this was already happening? Well, for one thing it was not at all clear before the Mongols arrived on the scene that these ideas would prevail. Ghazali was fired from his teaching position for pushing his theological (as opposed to legal) views on his students. And Ibn Rushd in the 12th century published what might be the best titled reply in all of history when he wrote The Incoherence of 'The Incoherence'.

One of the reasons why this can be difficult to write about is that even this can be overstated. Ibn Rushd as the last major philosopher of classical Islam himself fell out of favor for his views, and lived in the extreme periphery of Islamdom.

So a lot of this history gets muddy, after all it's very difficult to write a history of how an entire religion or culture reached a broad consensus on adopting an intellectual position vis a vis outside influence, but the Mongols are often written as being critical. By creating a century of utter chaos in the intellectual center of Islamdom, you go from a period where Islam has a vibrant intellectual life in the 12th century to an almost total retreat into existing religious law and rejection of scholasticism or attempts to merge philosophy and religion.

In other words, as a famous phrase in the field goes, following the Mongol conquests the "Gates of Ijtihad [individual reasoning]" were firmly closed.

A critical figure in this transition was Ibn Taymiyyah in the 13th and 14th centuries. Drawing on the work of Al Ghazali, Taymiyyah espoused an ultra conservative ("inward" as you might say) form of Islam that totally rejected not just philosophy but the entire field of theology as well. As with Ghazali I think this can easily be misunderstood, because today we might term all of this as "theology." Rather, Taymiyyah and Ghazali rejected Islamic religious philosophy as in their view it had no basis in the traditions or the Quran. So the kinds of theological questions being asked, for instance, by the Scholastics like Aquinas were rejected as innovation and instead Taymiyyah called for a return to and reliance on the internal traditions of Islam. I think you could make the case for Ibn Taymayyiah and Al Ghazali between them being the most influential "philosophers," though of course they would utterly reject that term, in Islamic history.

So the TL;DR is Islamic empires did not become isolationist after the Mongols, but there's a case to be made that Islamic thinking did.