How did so many Muslims in The Philippines support Duterte?

Because for the first time the Filipino Muslims have been recognise and not being alienated among others. This group of people are begging for help for a long time and no one dare to do something, they are hearing them but they don't listen. The politicians always concentrated in Luzon where the centre of big population of Philippine is. Duterte brakes the pattern because he recognised the issues. He doesn't see them as a Muslims but they are Filipinos who deserve being serve by public servant.

You ask "How did so many Muslims in The Philippines support Duterte?"
I answer by voting for him. He is doing what he said he will do if he us elected

Maybe because he in the only pne from Mindanao who run for presidency.

Well, they were probably fed up of all the corruption,war, violence crime and drugs and politicians who re only there for themselves, the other candidates didn't appeal to them. Many felt just like the non-muslims that what the Philippines need is a tough no-bullshit president who can stand up for all of Philippines, and probably saw what he did as Mayor of Davao and how he changed them.

So to finish answering your question, they went to the polling station, voted Duterte, went home and hoped he won....the next day they were happy because he won.

Because they are anti-crime and drugs