The basic answer is the Islamic conquests, followed by ~1300 years of Islamic rule.

Raids into Palestine began under the Prophet Muhammad (albeit not very successfully) but the entire region came under Arab control following the Battle of Yarmouk in 636 and continued to be ruled by Islamic empires with only minor interruptions-- like those nagging little crusades-- until the British occupied Palestine in WWI.

Now, it was not the conquests themselves that caused the conversion of the local populace. Indeed, non-Arabs couldn't convert in early Islam unless they joined an Arab tribe. That being said, quite a lot of Arabs were likely living in Palestine as nomads or semi-Nomads prior to the Arab conquests and even more likely moved in and settled following the conquests.

One of the challenges in discussing this issue, however, is that we don't have census data on a decade by decade, or even century by century, basis, and therefore it's difficult to say when or necessarily why conversion took place. What is clear is that forced conversion (the old "spread by the sword" canard) didn't happen except in very unusual circumstances.

Rather, and this is true of all of the conquered territories, not just Palestine. There was a process by which conversion to Islam was made attractive to the non-Muslim populace such that conversion occurred on a gradual basis on a bell-shaped over the course of several centuries.

There were multiple factors involved in this, and the reforms of Abd al-Malik in 7th/8th century seem to be important. For one thing, the construction of religious buildings like the Dome of the Rock appear to have been a direct propaganda appeal to non-Muslims. The Quranic verses in the Dome of the Rock appear to be chosen specifically for their propaganda value in conversion, for instance:

S In the name of God, the Merciful the Compassionate. There is no god but God. He is One. He has no associate. Say: He is God, the One! God, the eternally Besought of all! He begetteth not nor was begotten. And there is none comparable unto Him. Muḥammad is the Messenger of God, the blessing of God be on him.

And so on, which you can read here.

He also Arabized and Islamized both the coinage and, perhaps more importantly, the bureaucracy. Where before non-Muslims carried over as the dominant force in administration in Greater Syria, from that point forward being an Arabic-speaking Muslim would be a definite career benefit. There were also negative impacts associated with being a non-Muslim, most famously the paying of the Jizya poll tax. That being said, I think the impact of the Jizya can be overstated as it wasn't uniformly applied and tax benefits of converting weren't always clear.

In any event, through these economic, social, and political factors, Islam eventually became the dominant religion in the whole of the Middle East.

Source wise:

On conversion rates and practices, see Richard Bulliett's monograph Conversion to Islam in the Medieval Period. It's quite old but the method and analysis hasn't really been improved on. It also discusses why it's so hard to know precisely when conversion took place and some approaches to dealing with that.

For the progress of the conquests and the early Islamic state, I usually recommend Robert Hoyland's In God's Path as a solid and up-to-date introduction.

edit: typo.

Palestine was conquered by the Islamic Caliphate beginning in 634CE, but the population remained mostly christian until Saladins conquest in 1187. The decline of the Byzantine/Roman Empire after the Fourth Crusade reduced christian influence even further. Followed by other muslim powers like the Mamluks and the Ottoman Empire later one. And as special taxes were imposed on non-muslims only, the new local rulers - which were mostly muslims - had no interests in converting their population. But it has to be noted that the Samarian people in Palestine of that time seem to be only historical case where we have evidence of mass conversation to islam. This didn't affect the majority of the population though.

The change came slowly, and accelerated or was even halted when the ruling class changed. But it can be generally said that it increased with the Mamluks and Ottomans.