How and where did Islam spread after Muhammad's death?

Identity.

Jonathan P. Berkey, The Formation of Islam: Religion and Society in the Near East, 600-1800, 2003

The conquest of Arabia

After the death of Muhammad in 632 CE, the young Muslim federation came under strain. Some of the tribes decided that as their loyalty to Islam had been primarily to Muhammad himself, his death allowed them to end their allegiance to Mecca and to Islam.

To make things more difficult, the Prophet had not left clear instructions as to who should lead the community after his death.

Fortunately the community immediately chose the Prophet's close companion and father-in-Law Abu Bakr, as his successor. Abu Bakr was known as the first caliph (from khalifa, the Arabic for successor).

Abu Bakr took swift military action against the communities that wanted to break away. These campaigns, known as the apostasy or ridda wars, effectively consolidated Arabia into a single country under Muslim control within two years