Not by sword alone
When one looks at Near Eastern history one of the amazing things is the rapid victory of invading Muslims and their religion over Christian Byzantine empire. In only a few hundred years the dominant religion, Christianity, had all but been wiped out and not only the invading Arab population from the East but also majority of the natives in Levant and North Africa became followers of Islam.
Contrary to popular belief, Muslims did not forcefully convert the Christians living in the occupied lands. Christians had prevailed through the most horrific physical persecution under Roman Empire and only got stronger from the martyrdom of thousands and thousands. Islamic sword was not the one that brought victory to the new religion.
Benevolent rulers
In fact, there are many memories of just Muslim rulers who protected Christian shrines and people during the times of the Four Caliphs, especially Omar, and under the Umayyad Caliphate under which Islamic armies swept towards East and West with little resistance. It was only later, during the Abbasids and Fatimids, that the oppression of Christian population became widespread in the Islamic Empire.
Christians' deep hatred of each other
So if not really by force of the sword how followers of Muhammad took over the cradle of Christianity where the religion was born and spread during six centuries?
Well, in the beginning it was largely the decision of the Christians themselves to adopt the new religion. There were two major factors pushing people from churches to mosques.
First, the extremely bitter infighting between various Christian denominations and churches. In the East it culminated on the battles between groups that used different terminology to describe who Christ is. The Chalcedonians had power and more often support of the Byzantine emperors then the pre-Chalcedonians often called Monophysites. For example in Egypt, these quarrels led to expulsions, outright killings and such a harsh Constantinople's rule over local Christians that Coptic Christians gladly opened the doors of Alexandria to the invading Islamic army greeting them as saviours from Byzantine terror.
Another example was the quarrel between native North-African Christians, mostly berbers, and the Latin church. It was all about a schism, dispute of the right of lapsed Christians to hold high positions in the church hierarchy after the Roman persecution was over. This continued for centuries and created the Donatist church persecuted by Rome. Islam found many converts from this church which disappears when the men of East occupied North-Africa on their way to Spain.
Money money, money
Secondly, money. This seems to have been the major factor during the glory days of the Umayyad Caliphate stretching from Atlantic to Hindus. Islamic rulers set special taxes to Christians from which Muslims were free. As a consequence of this simple policy long lines of citizens were seeking to convert to the new religion in order to get the tax relief and other material benefits the Islamic Empire gave to its Muslim citizens.
So also today
Accordingly, as Christians we should not blame only Muslims for the loss of Levant and North Africa to Islam. Rather, with certain sadness and willingness to learn from past mistakes we should consider the same factors today when secularism offers freedom from Church taxes and payments, Christian quarrels alienate many good people and Islam arrives to Europe with rock solid faith in their teachings and divine justice in their manners and traditions.