Short version? Magic.
Slightly longer version? From the power of their conviction.
So, if that is the case, are insane people in the setting invincible? Because no one has more perfect conviction than insane and delusional people. If just conviction turns into magic, why doesn't it work for everyone and in particular why doesn't it work really, really, REALLY well for the completely deluded?
What I am beating around the bush about here is that however you want it to work, it has to be internally coherent. You can't just willy-nilly take a religious argument about Faith, and apply it to a situation where the setting has completely changed. Faith isn't powerful because it's conviction, or at least, the people who believe that Faith is a virtue don't believe it is powerful just because stubbornness is a super-power. They believe that the power of faith comes from what you believe in.
Back in ancient times when people took oaths, they didn't believe that the oath had any particular power. The power the oath had came from whom or what witnessed the oath, and how that power would respond to it. That's why when they took an oath, they swore by or on something, because in doing so they were attracting the attention of something that would hold you to the oath or empower you to keep. That's true of pretty much any oath making culture.
So if you remove the culture and the setting things that make oaths and faith and whatever have meaning, you need to look around for some other answer than 'oaths' or 'faith'.
So even in a generic D&D campaign, the power doesn't necessarily come from the gods.
Sure, but what do you mean by 'gods'? Because if your setting has nature spirits, or ancestral powers, or primal cosmic forces and swearing an oath on them actually draws upon power, those things are the gods of that setting. And as evidence, I point to all the vast numbers of people throughout human history who when they thought about the idea 'gods' pointed to those things.
Again, what are you actually saying is going missing when you say, "There are no gods." Tell me what that is like, and then maybe I can brainstorm up some help.