Language and legacy are both critical elements.

Islam first appears as a religion and a political power in the seventh century and rockets to prominence within decades. Within only a few decades, Islamic rulers were able to capture territory from the Maghrib (North Africa) to Persia to Anatolia. Out of nowhere, this new religion captures huge swaths of the territories of two enormous empires, the Romans (Byzantines) and the Sassanids (Persians).

In the years immediately following these conquests institutions are left largely untouched. Many former Roman officials continued in their posts. Considering the youth of the Muslim polities, the bureaucratic base just wasn't there. They weren't used to ruling over such an enormous populace and land base. Given their official positions of tolerance for Jews, Christians, and Zoroastrians, it seemed natural to allow them to continue the work they'd done before. This policy had a broad reach. Some early Muslim currencies were even repurposed Greek coins that might have been stamped with Arabic phrases; slightly later ones would emulate Greek iconic styles and depict the ruler. This meant that within the body of the new Muslim territories you had access to a native Greek bureaucracy, native Greek speakers, native Greek texts-- the same for Persian territories, which allowed for the continuity of local rulership there, usually followed by slow conversion to Islam.

In the ninth century the caliph al-Ma'mun began pushing for a widespread conversion to the Arabic language. Records and texts were translated to Arabic and allowed the language to play a similar role that Latin did in Western Europe at the same time, or the one that Greek had in the Mediterranean during the post-Alexander period and indeed up to the time of the rise of Islam.

Greek proficiency in Western Europe didn't really gain traction until the late Medieval period, when the expansion of groups like the Seljuk Turks and the Ottomans shrank the size of the Roman Empire under the Greeks and struck the fear of God into them. While monks might aspire to read and translate some Classics, there weren't enough Greek teachers to go around. While they might have preserved Greek texts, that doesn't mean they were capable of reading them. On the other hand, incredibly influential scholars like al-Khwarizmi were able to develop ideas of math and science thanks to their place in a wide-ranging intellectual world. Al-Khwarizmi wasn't limited to native Persian texts or even native Arabic ones; translations were easily accessible to him. In short, Muslim scholars didn't have to start from scratch. They were able to look at the historical foundations of their field and grow those fields based on that education.

The more unified rule of the early centuries of Islamic rule also helped. While Western Christian polities were divided internally and had limits on inland international travel, travel along the Mediterranean was always common, and the common rule across the Islamic world in the early centuries allowed for the easy spread of information, texts, and thought.

When the Western world began to benefit from the fruits of those labours-- through access to new or classical texts via increased regional trade or spoils from the Crusades, or through the arrival of Greek scholars like Bessarion-- they did much the same. Christian theologians like Thomas Aquinas even referred to Muslim scholars like ibn Sina (Avicenna) and Algazel (al-Ghazali); when he refers to Jewish scholars, they include Maimonides, whose work has a unique place in the Muslim world that would be its own multiparagraph answer to elaborate upon. The Christian apologian Petrus Alfonsi was born a Jew in Islamicate Spain and benefited from its intellectual culture before his conversion, then used those arts for the benefit of his new religion.