I think it ultimately boils down to persuasion and the environment. When you're surrounded by 'faith', you can pin a lot of things onto that -- much like how some folks buy into horoscopes, homeopathy, statues, prayer cards, etc.
I know someone who considers themselves to be really lucky because they win a lot of contests -- but they only win a lot of contests because they enter a lot of contests.
I used to have warts on my hands and feet when I was a kid. It wasn't bad, only one on a finger and two in a benign spot on my feet. But they were very persistent for many years. I never saw a faith healer, but they still fell off on their own one day, never to return again. I had actually completely forgotten about that until I read your message, likely because it held no significance for me. The coincidence of faith healer + warts falling off appears more significant that warts falling off with no apparent instigator.
I'm not trying to suggest you're lying or anything, but these kinds of stories get warped in our memories over time and sort of settle on a more significant, but less accurate version.
Personally, I think that these things seem real to people because they want to believe it's real, and so they'll latch on to anything that remotely stands out and attach a meaning to it. Over time their memories of the event settle on that idealized version of it.