The story probably isn't true. There's some debate about the validity of the science. But that doesn't matter. Chances are when you saw the word "epiphany," you thought of that Greek guy who made a great discovery in his bathtub and ran naked through the streets shouting "EUREKA!"
(For the record, his name was Archimedes. He was a mathematician who was given the job of finding out whether the king's crown was pure gold. His insight was that, just as his body displaced the bathwater to reveal its true weight, so would a crown dunked in water.)
Now Archimedes figured out plenty of other things––including an approximate value for pi. But we remember him best for that Eureka moment. Why?
Epiphany means a revelatory manifestation. An inspiration so vivid, so amazing that you feel like you're in the presence of the divine. An idea that transports you and makes you feel like running naked through the streets.
Of course, having the actual insight is crucial. As Deborah Lytton pointed out in her post, our characters need to have those revelations too. But if we want to give those moments maximum impact. If we want people to remember them, then we need to make them manifest. Make them transport us from our warm baths.
Then we can cry, Eureka! Because we really will have found it.