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.