Islam never collapsed. I take it by this you are referring to that video of Neil deGrasse Tyson's where he blames the collapse of Islamic science on Al Ghazali. One should keep in mind that Tyson is a scientist but he is not a historian of science, and while there is a touch of truth to his story of Islamic science, his facts are basically wrong. Ghazali, who was hugely influential, did not shift the entire Islamic world from being 'pro-science' to 'anti-science'. Answering your question requires discussing the history of Islamic philosophy and Islamic science, because Tyson has got the two completely mixed up (as will be explained soon). I will focus on the history of philosophy, but unfortunately I'm not well-acquainted with the history of Islamic science. The collapse of Islamic science may be attributed to the conquests of its intellectual centres, Baghdad and Cordoba, but this falls outside my knowledge and I don't want to spout generic lines about the Mongol sack of Baghdad.

In the first centuries of Islam, two rival philosophical schools of thought developed: the Mu'tazilites and the Ash'arites. The Mu'tazilites believed that reason can derive moral and judicial judgements which are in keeping with the divine (i.e. God's moral and judicial judgements). The Ash'arites believe that morality can be defined only by God, and reason cannot substitute the divine. An example Mu'tazilite position: Reason tells me that killing indiscriminately is wrong, so even if God did not decree that in the Quran, it would still be wrong, and as God's justice is logical it is implicitly immoral from a religious perspective. The Ash'arite would argue that killing is wrong because God judges it to be wrong; if God were to have decreed it morally right to kill indiscriminately, then it would be morally right regardless of what human reason might tell us.

Ghazali was an Ash'arite and wrote a treatise against the Mu'tazilite position, The Incoherence of the Philosophers, and the Mu'tazilite school died out in that time. Ghazali was also a hugely influential Sufi (mystic) and popularised Sufism; he is easily one of the most famous Muslim philosophers and his influence was widespread in Sunni Islam. Ghazali wasn't anti-reason, but he believed that philosophical positions should be derived from Islamic sources. Logic still had a place, but it was subordinated to the will of God (the Mu'tazilite position in effect made logic an equal to the divine). He engaged in the philosophical method to deconstruct it, and though he would have probably hated to be called one himself, he was a philosopher by virtue of his works and their influence.

So when Neil deGrasse Tyson talks about Ghazali as a crusader against science, he has in fact completely misunderstood and misrepresented the philosophical rivalry of the Mu'tazilites and Ash'arites. And there continued to be Muslim science after him: Ibn Rushd (Averroes), a Mu'tazilite, would write The Incoherence of the Incoherence as a rebuttal to Ghazali in the 12th century. Ibn Khaldun (1332-1406), widely considered the greatest polymath the Muslim world ever produced, was born over 200 years after Ghazali's death.

Edit: There are a couple of things I wanted to add to this but I ran out of time earlier. First is that the Mu'tazilite position never really died out, though it lost pretty much all its influence in the Sunni discourse. Shi'ite theology developed a similar distinction between the Akhbaris (whose position is roughly equivalent to the Ash'aris) and the 'Usulis (whose position is similar to the Mu'tazilites). In the case of the Shi'a, the pro-logic 'Usuli school has dominated since at least the 16th century and Akhbari philosophy is limited to a few small pockets of the Muslim world. There are also Muslim intellectuals and scholars today who are self-described or else may be described as neo-Mu'tazilites, including Abdulkarim Soroush and Khaled Abou El Fadl.