Algebra is just one of the sciences that comes to the West through the Islamic world. Another straightforward example can be seen from the origins of the word chemistry. Chemistry, of course, comes from al-khimiya, which becomes the famous art of alchemy when it enters into Latin usage. The "creator" of algebra was al-Khwarizmi, a Persian scholar whose name is the origin of our modern world algorythm. But how did this happen?

First we should probably look at the origins of the Islamic world as a political entity. The regions that came to be predominantly Muslim over the course of the medieval period were largely taken from two empires that were powerful in West Asia during the early Middle Ages, the Byzantines (or Eastern Roman Empire) and the Sasanians (one of the groups generally grouped in English under the name "Persian Empire"). In time, Muslim powers came to dominate former Byzantine holdings in Anatolia, the Levant, and North Africa, and the Muslim world was strongly influenced by the structure and narratives of the Byzantine Empire. Importantly, Islam has styled itself religiously as the successor religion to Christianity. This of course extended to being a political successor to the Byzantine world as well and could even repurpose some old Christian narratives about the tyranny and sinfulness of Rome to refer to the Byzantines, who called themselves Romaioi, or Romans. Anyhow, control over former Byzantine regions also allowed them access to the fruits of Greek learning, including the classic works of scholars like Plato and Aristotle. These works had largely been lost in the West as a result of declining understanding of Greek and the collapse of the Western Roman Empire. Some monasteries might have had these classic texts tucked away somewhere deep in the stacks, but in large part they were lost, and the more fragmented Western world (the Eastern and Muslim world was generally more connected in the early medieval period) had limited communication which made collaboration and the sharing of learning more difficult.

Scholars early in the Muslim era were just using the texts as they were originally written, in Greek, as Greek was the standard language of the Roman and Byzantine worlds all the way from the time of Plato to the expansion of the Muslim caliphates. In the early ninth century, a man called al-Ma'mun became the ruler of the Abbasid caliphate. His father had been Harun al'Rashid, one of the most famous and well-liked of the caliphs, and he achieved his throne after a war with his brother al-Amin. As a result, al-Ma'mun had to settle his empire and set himself apart from his illustrious predecessors. One way in which he did this was through the promotion of Arabic and sciences in general in the Muslim world. Before this, the Muslim empires had sort of had a "business as usual" policy when it came to language. If the locals used Greek or Syriac in their administration before their conquest, they continued using it, as that was what scholars and bureaucrats were used to. Under al-Ma'mun people began moving toward Arabic generally, which further helped to promote communication as it was a common language throughout the Muslim world, and could be used in a way that Latin was throughout the Western world for the medieval period and beyond. While some of al-Ma'mun's efforts to change the culture and practices of his empire were largely rejected, the transition to Arabic clearly worked, as Arabic is one of the most common languages in the world today.

These texts finally came to the West through a variety of methods. Most obvious was the larger connections between the Islamic world and the West as a result of the Crusades. Westerners who entered into the Near East could return with plunder including books, Islamic art, products, etc., etc. These could then be translated by Latin Christian monks who had lost the Greek classics that formed the bedrock of the Islamic sciences. Another translation movement in the West was later caused by the shrinking of the Byzantine Empire as a result of the expansion of the Seljuks and, later, the Ottomans. Scholars from the Greek world like John Bessarion (who has my personal favourite name in all of my medieval history readings) helped to translate and to spread understanding of Greek in the West.

As for specific inventions, Muslim and Arabic scholars innovated in all sorts of fields. Mathematics were radically changed by the world of scholars like al-Khwarizmi, who you can, as you note, blame for any rough algebra lessons in school. Doctors like Ibn Sina (known in the West as Avicenna) compiled and improved upon the field of medicine. The foundations for inventions like glasses and telescopes were in the Muslim world with scholars like ibn al-Haytham, which led to innovations in astronomy (and astrology). The traditions of philosophy that Muslims so admired in the Greek world were continued by thinkers like ibn Farabi, ibn Rushd (Averroes), and al-Ghazali, whose work became important to Western scholars like Thomas Aquinas in their own innovations.

Citations:

Gutas, Dimitri. 1998. Greek Thought, Arabic Culture: The Graeco-Arabic Translation Movement in Baghdad and Early ʻAbbāsid Society (2nd-4th/8th-10th Centuries). Routledge.