This question seems to be more religious rather than historical, so it's difficult to give solid reasons without dabbling into religion itself. What we do know is that Abrahamic religions came into contact with pagan beliefs, they invariable won. Both started in places of distinct polytheism (Christianity in the greater Roman Empire, Islam in tribal Arabia) and converted the local populations before moving forward.

In addition, both are proselytic religions while polytheistic beliefs are almost always ethnic. Basically this means that Islam and Christianity actively seek converts and desire to spread their religion to all corners of the map. You can compare this to polytheistic beliefs which are deep rooted in their culture and don't care about converting the guy 1000 miles away. Think of it like this, a Muslim would love to convert someone in say, Persia, even if they're at war with the country. At the same time, someone following the Greek mythological religion couldn't really care less about converting someone so far away, he just cares about his own beliefs. Beyond that, ethnic religions tend to be very hard to adopt, since they require culture to function. Proselytic religions like Islam and Christianity are not really cultural beliefs; at their most basic roots, the religions can function anywhere and do in fact exist in many different cultures across the world. Islam especially has a strong unifying factor which encourages believers to care more about religious ties to the ummah, or community, rather than pure ethnicity or tribal background.

Finally, you have the existence of zealotry. Unlike other religions, Christians and Muslims have sent their best people out into the wild to convert their neighbors. During the early years of Christianity, zealotry and staunch belief was what really prevented Roman oppression from being effective. People were willing to hold to their beliefs and the most devout amongst them were wholly willing to die for their cause. For some examples of this, you can look at the early martyrs of Christianity. This doubles for Islam, whose believers also faced the same kind of external threats. Muslim converts would happily contribute to the army and were also willing to die as martyrs. Because they can convert people, this zealotry is reflected in newer converts in new areas, meaning that people of faith who are in the minority somewhere will actively fight to gain followers. Eventually, because of the appeal of monotheism, these minorities win over the population.

Mainly both religions Islam and Christianity are very willing to have converts, unlike other religions who are very ethnocentric, like Judaism. I can't say much about Christianity but I can about Islam.

After the death of Muhammad, there were four Caliphs, known as the Four Rightly Guided Caliphs. All the caliphs after that were considered to either a) not follow the religion almost perfectly b) the people didn't follow the religion almost perfectly. There were battles and swords were indeed drawn, but it wasn't forced or anything. After conquering land, Muslims did not plunder it and did not force people to convert to Islam. They didn't have to rape women or kill children to show that they were in power. Instead, they said that they were the rulers and and all they had to do is a pay a tax (jizya), but this concept wasn't new to them at all. They had to the same to the previous people, whether they were Persians or Byzanitines, etc. They were allowed to follow their own faith and keep theri own courts of law and have their own houses of worships. For a start, since the Muslims were so kind and tolerant with other religions, this was the first step of people getting interested in Islam at that time. After that, it comes to religious aspect like the Quran, Sharia, etc. Islam grew back then, and still is because its teaching appeal to a wide range appeal, just like Christianity.