Islam spread quickly because the Islamic state was initially able to expand very quickly. The Middle East at the time was ruled by the Byzantine Empire and the Sassanid Persians, who had exhausted each other fighting a series of devastating wars that left them with no real capacity for defense when the united Arab Muslims arrived in force. The Byzantines were also ruthless enforcers of Christian orthodoxy. "Heretical" Christianities like Nestorianism had a large following in Egypt, Palestine, and Syria, and these populations welcomed the Muslims as liberators, often collaborating with them.

Religion typically follows empire, not necessarilly because of forced conversion, but because it allows cultural exchange through trade, colonization, and the presence of garrisons in the cities. Empires also support and legitimate their state religion, giving it a stronger public presence and creating social and economic incentives to convert, such as the jizya tax on non-Muslims. Christianity heavily benefited from these factors, it just used Rome's existing imperial structure instead of building its own.

Later, like other religions, Islam spread quickly through trade along the Silk Road in Central Asia and the sea routes in the Indian Ocean and SE Asia.

The spread of Islam spanned from Morocco to Indonesia, so it's difficult to say there was one overarching theme that characterized the whole thing.

Generally, early on in Islamic history, Muslim armies had the upper hand on the battlefield, despite their numerical inferiority. The Byzantine and Persian Empires had been in a state of almost constant war for decades, and when Islam arrived in Arabia, it was in a perfect position to take advantage of their military and political weakness. Thus, but the 630s, Egypt, Syria, and much of Iran were conquered by the caliphate led by Umar ibn al-Khattab.

The treaties that the conquering Muslims and the provincial governors came to usually included provisions for freedom of religion for non-Muslims living under Muslim rule. The "spread by the sword" argument applies to Muslim political control, but not actual Islamic belief. In fact, according to many estimates, by the mid-700s, over 100 years after the conquest, only about 5% of the empire actually believed in Islam, the rest being Christians, Jews, and Zoroastrians.

After that initial expansion of Muslim political control in the 600s and 700s, the military spread of Islam slowed. Instead, Islam spread to regions such as West and East Africa, China, and SE Asia mostly through trade. Kings in those regions regularly would convert to Islam to take advantage of the economic benefits that came with being part of the vast Muslim world. It would generally take generations for Islam to spread throughout the local populations, and if you go to some rural regions of Africa or Southeast Asia, you'd find many people still following traditional religions, as the "Islamization" of those regions is still ongoing.

Sources:

The Great Arab Conquests by Hugh Kennedy The Venture of Islam by Marshall Hodgson Lost Islamic History