There are a couple of narratives that attempt to explain this. The traditional narrative proposed by later Muslim scholars is that when the Abbasid dynasty usurped power from the Umayyads in 750 CE, this represented a shift in power from Arabs to the numerically and culturally much stronger Persians. From this, we have two dovetailing explanations of the interest in science: 1) they wished to recover the former glory of the Persian Empire, and 2) in order to gain political legitimacy, they essentially drew together all the various scholarly traditions from their polyglot empire, had everything translated into Arabic and established schools of translation and development, where works of classical Greek, Babylonian, Egyptian and Persian science and philosophy were built upon by succeeding generations of scholars. A proper ruler is one who has a court of scholars, became the principle, so when the caliphate began to fragment, the petty kings all drew to themselves their own scholars in order to seem more kingly. Scholarship itself was something of a meritocracy (for men): even a village lad, if sufficiently able, could be essentially passed up a food chain of scholars and eventually find a position at a court. These trends reinforced each other, plus scholarship had a lot of practical value and applications. The quest for knowledge was supported by some cherry-picked quotations from the Qur'an and Hadith in order to make it appear that a proper Muslim ruler had an obligation to support scholarship. This period is held to have ended (again, this is a traditional narrative) around the year 1100 as the notion that a proper Muslim eschewed the natural world in favor of prayer and meditation began to take hold.

But there are some real problems with this traditional narrative. For one thing, the idea that scientific development came to a halt is provably false: it continued, albeit at not quite such an increasing pace, well into the 16th century CE and only then really began to stagnate. For another (and forgive me, I'm on holiday and don't have sources with me, but the scholar whose work I'm badly summarizing here is George Saliba), the way in which the "translation movement" sprang into print in full bloom is unpersuasive: for the initial generation of works in Arabic to have been so sophisticated implies at least another generation of unseen scholars beforehand, and therefore the explanation that the early Abbasids were the driving force is problematic. There's some real driving force(s) for scholarship for its own sake during Umayyad times, one(s) we don't know much about because we don't know nearly as much about the Umayyads as we'd like.