The answer to the question above would be that the root in egomania is the following one: Ego. This Latin root means "I".
Everybody knows that someone with a big ego is someone who is always talking about himself and tells everybody about how great he is because he is always thinking about "I", that is, about himself. What is more, "mania" refers to "a very strong interest in something that fills a person's mind or uses up all their time". As you can see, when you put together ego+mania, you get the word with the definition stated in the example.
