Why is Roman Catholicism the dominant religion in the Philippines?

It's because Spain colonized the Philippines for 333 years (1565-1898).

There are 3 reasons/motivations for colonization, it's for Gold, God, and Glory (3Gs). For those 333 years, Spain used the Philippines' resources to make trade not just for their own country but to others as well (Gold). Spain at that time is already a Catholic country, so they see it as their mission to convert others into Catholicism as well (God). It is important to take note that Philippines already has their own "religion" at that time, and Spain managed to convert them either way. So in 333 years, they were able to convert almost all Filipinos due to their power (Philippines is in no way as advanced as Spain that time (i. e., lack of weapons etc.)). By colonizing other countries, Spain earned fame and power (Glory).

The Spanish controlled the Philippines for hundreds of years before losing it in the Spanish-American War. As was always the case, once Spain colonized or controlled a region, Catholic missionaries would arrive and evangelize the indigenous population. Thus, the Catholic Church was able to win many converts to Christianity. As a result, the dominant religion in the Philippines is Catholicism.