Why did the father of algebra speak Arabic instead of Persian?

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The language of science at the time was Arabic, just like now a days a lot of people from around the world write their articles and books in English even though that might not be their main language.

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Arabic was the primary literary language in the Muslim world at the time. Partly it's because the early caliphate (the center of power) was Arab, but another reason was that Arabic was the only language in which the Quran was supposed to be read—unlike in Christianity where the Latin version of the Bible was also considered a valid Bible. Therefore, throughout the Islamic world, literate people learned to write and read Arabic, even after the central power of the caliphate waned. This made it an ideal literary language. You can also think of it like how writers today tend to publish papers in English even though it's not their first language.

In much of the Islamic world, Arabic duly became and remained the primary language. Even in places that held on to their local tongue, like Persia, Arabic became an important and even dominant language for writing literature, including philosophy. This is why philosophers from Persia and central Asia—including no less a thinker than Avicenna—wrote in Arabic, which was not necessarily their native language.

-- Peter Adamson (2016), Philosophy in the Islamic World p.4.

The use of Arabic was not restricted to Muslims either. Jewish author in the Islamic world, e.g. Saadia Gaon (d. 942) and Maimonides (12th century) wrote extensively in Arabic.

Further reading

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Persian is used synonymously with Iranian, but it is not a helpful generalization. He was actually Khwarazmian. Khwarazem was a distinct region in Central Asia, south of the Aral Sea, with its own language. It was one of the most distant places that the Umayyads conquered. Because of the distance, the Arab ethnarchy was no so rigid, but instead they intermarried with Khwarazem families, and certain Khwarazmians remained influential. The result of this interaction was a culture that created the Abbasid revolution (750), which led to the proliferation of Khwarazmians in the Caliphate such as Al-Kwharizmi.

The Umayyads (661-750) went through great lengths to destroy the Khwarezmian script. Despite this, the Kwharazem language persisted in the region until the Mongol conquests, being written in a form of Arabic.

Al-Khwarizmi lived and worked in Baghdad, which would remove any possibility of him writing in the Khwarazmian. Regardless, if you wanted to reach the greater world, you wrote in Arabic. The revival of the Persian language in Central Asia began during Iranian Intermezzo (821-979). Still, Arabic remained the Lingua Franca along the Silk Road. Al-Biruni (d. 1050) wrote almost entirely in Arabic.

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