It's kinda like the Western equivalent of how many of the tropes that make anime anime come from kabuki and other traditional forms of Japanese theater.
Try reading about and watching (video of) kabuki. Typical plots involve the standard hot-blooded hero with a strong sense of justice exacting vengeance on a villainous rival. Kabuki actors deliver their lines in an over-the-top fashion, strike cool poses as they're introduced, and even undergo "transformation sequences" that make important revelations and up the stakes in the plot -- much like anime characters. (Transformations in kabuki are effected with rapid costume changes and traditional stage effects.) Like anime, kabuki is also recognizable by being highly stylized, with realism de-emphasized in favor of spectacle and looking cool.
Anime tends to share with Japanese theater a much more lingering emphasis on the atmosphere of the setting and the emotional states and thoughts of the characters compared to fast-paced, exposition-heavy Western media.
Of course there are plenty of exceptions, and some anime is even explicitly styled after Western media.
