Thank you for your question DisWastingMyTime.

It helps to understand where the word "Arab" comes from. As you may know, words in the Semitic branch of the Afro-Asiatic family tree stem from triconsonantal roots. The triconsonantal root from which the word "Arab" is derived is 3 - R - B (ع - ر - ب). Now, the distinction between 3 (ع) and gh (غ) is theoretically present in the last common ancestor of both Arabic and Canaanite dialects/languages, but was only preserved in the later Iron Age by the Arabic language. 3 and gh merged into a single phoneme: 3 (ع), a voiced pharyngeal fricative. The word for "west" is gharb, or gh - R - B if the root consonants are isolated. The word for the Arabs in Arabic is 3arab, or 3 - R - B if the root consonants are isolated. Notice how the phonemes 3 and gh that we established and know with reasonable certainty have merged into a 3 in Canaanite make it so that the root for both "West" and "Arab" is 3 - R - B.

What I just explained to you earlier is an account for the possible etymology for the word "Arab." It's West. But west of what? Historians and those that dabble in the role linguistics can help us in making determinations about history think that the word "Arab" was attributed to the Semitic nomads/Bedouins that called the deserts west of the Euphrates their home.

This is partly why the city of Palmyra in the center of the modern state of Syria has strong Arabic and Aramaic roots. Which is why any discussion of the roots of Empress Zenobia of the Palmyrene Empire more or less boils down to her having both Arabic AND Aramaic ancestry, given how both of these peoples contributed to Palmyra's populace. I digress.

The "locals" of the Levant in general (or Greater Syria, depending on the term you prefer) were the Aramean peoples. Aramaic was a dialect continuum that spanned the modern states of Syria, Lebanon, Jordan, and Palestine, in addition to significant swathes of land in southern Turkey on the border with Syria. The "locals" of the Levant/Syria were the Aramean peoples who spoke their traditional Aramean dialects.

HOWEVER, a few short centuries prior to the Islamic conquest of Syria, the Ghassanids from Yemen immigrated to southern Syria where they were centered around the city of Damascus and the city of Bosra, more or less correlating with the modern governorates of Daraa, As-Suwayda, and Rif Dimashq, in addition to the arid lands east of the river Jordan (modern Kingdom of Jordan). The Ghassanids were a client state of the Eastern Roman Empire (the Byzantines as we refer to them today) and were allowed to somewhat govern these lands under the larger umbrella of Constantinople. The Eastern Romans used them as a "buffer" zone between them and the lawless Bedouin raiders to the south in the Arabian Peninsula.

So, Arabs HAVE a well documented history in Syria prior to the Islamic conquests, particularly around Damascus, Bosra, Palmyra, and the general area of Jordan.

So, how did the Syrians go from speaking a diverse plethora of Aramaic dialects to speaking a dialect of Arabic? Why do Indians learn and speak English today? Because of the influence of the British empire. The Aramaic speakers learned and adopted Arabic because it was the prestige language of the new Empire that has come to govern them, the Arab Islamic Caliphate. As time pressed on, not only had the locals adopted the language, they had also adopted the "Arab" ethnolinguistic identity.

Relations with the locals were generally good. The Arab Islamic Caliphate expanded so rapidly and successfully because they were excellent military tacticians and excellent governors who understood how to rule over those who did not share their new religious belief. The local Christians were permitted to continue to conduct their private and community affairs with their own local laws and customs. The Islamic "state/government" imposed a "Jizya" tax on the non-Muslim populace in exchange for the military protection of the "state/government" from external aggressors.

Now, as soon as the locals started adopting the Islamic faith, some rulers continued to impose the "Jizya" tax upon them because they were still deemed "non-Arabs," even though this was clearly a violation of their Prophet's clear and concise instruction in the sermon he gave in his Farewell Address regarding the equality of all Muslims in the eyes of their Lord irrespective of whether they're Arabs or otherwise, and whether they are of darker skin or fairer skin. So even though relations were good as a whole, there remained tension from time to time regardless.

Furthermore, do keep in mind that there WERE Arab immigrants in the Levant following the Islamic conquests, just no where near enough to supplant the existing Levantine/Aramaic populace. That's largely why Syrians today tend to have distinctive facial features and fair skin, as opposed to the Arabs of the Arabic Peninsula who possess their own distinctive facial characteristics and generally a darker olive/tan skin tone (there is MASSIVE variability in both populations however; I merely describe the commonest/stereotypical phenotypes you come across).

"Arab" became the dominant identity in the Levant in the late middle Ages. That's also around the time when Muslims became roughly 40 to 50% of the religious adherents of the Levant as well.

I hope this answers your question.

Edits: Spelling and syntax.