"Epiphany" is actually a word from theology, from two Greek roots, referring to the moment when a god becomes visible. That is, the moment when somebody realizes, suddenly and intensely, that something is divine. In Christianity, "the Epiphany" refers to the moment, in the New Testament story of the birth of Jesus, when the Wise Men visited Jesus, and became aware that the baby child was the Messiah they had been looking for. By extension, an "epiphany" is a sudden, intense insight into something deep and important. The article says "In 1974, Pepperberg watched two programs on PBS, one about Washoe and one about dolphin and whale intelligence. 'And it sort of clicked for me that this was real science, that you could be a scientist and study this,' she said." What she calls a "click" was what the writer of the article calls "an epiphany." Her sudden realization that studying animal behavior could be "real science" was "an epiphany."

It is quite apparent that your English is extremely advanced! Epiphany means revelation, new personal discovery, or new idea. In this case, the "epiphany" is referring to her discovering that she had other interests aside from Theoretical Chemistry. That other discovery is bird research. Here is an example sentence similar to the previous one: After my epiphany that my bike is slow because of my low tire pressure, I went to put more air in my tires. Here, I realized that it was the air in my tires (epiphany) that was causing me to go slow. Hope this helps!

"Epiphany" means a sudden realisation of something, or the act of having learnt something (quickly).