How would I write a paper on why same-sex marriage should legalized in the Philippines?

I'm afraid I can't write your paper for you, but I can point you in the right direction. The first thing you need to do is provide a bit of political background to the issue. You might like to mention how, in recent years, efforts have been made in the Filipino Senate and Congress to legalize same-sex marriages, all of which have failed.
You could then go on to discuss the various somersaults on the issue performed by President Duterte, who has changed his mind on the issue on a number of occasions but now seems to support the legalization of same-sex civil unions, if not marriages. This was the government's reaction to the Supreme Court's early deliberations in the case of Falcis III v Civil Registrar-General (2018), which considered the constitutionality of the Filipino Family Code's insistence that marriage can only be between a man and a woman.
Although same-sex marriages remain illegal in the country, the Philippines has the reputation of being one of the most gay-friendly nations in Asia, with almost three quarters of those questioned by the Pew Research Center showing positive attitudes towards homosexuality. You could use this information to back up your argument in support of legalizing same-sex marriages. If the vast majority of Filipinos have no problem accepting homosexual relationships, then why not make same-sex marriages legal?
The human rights angle is another one you might like to consider. LGBT activists in the Philippines argue that same-sex marriage is a human rights issue and that gay couples have as much right to get married as their heterosexual counterparts. The Filipino government is a signatory to the 2014 UN Human Rights Council resolution to fight violence and discrimination based on sexual orientation and gender identity. And so, the argument runs, what better way to make good on this commitment than by tackling discrimination in marriage?
https://www.pewresearch.org/global/2013/06/04/the-global-divide-on-homosexuality/