Movies, TV series, books...stories are interesting on many levels. The plot. The cool visuals depicted in visual mediums. The cool visuals easily create-able in one's mind for written mediums. The writing style. How dialog and subtle actions are used to sketch out the personalities of different characters. The stereotypes invoked. How the characters compare to various archetypes. For a surprisingly large number of pieces, the (bad) attitudes to parenting and children. The cultural memes portrayed, intentionally or not. The hidden and not-so-hidden messages and propaganda.
The viewer/reader can analyse the morality of each situation and its resolution, and compare to what happens and the results. Can pick up the relationship mistakes the characters make, and consider what would have worked better. Can observe the characters and their personalities, because, well people and life are interesting.
And, of course, there is the sheer enjoyment of it all. How cool is it to see a cyborg, covert-ops agent go roof-jumping, then charge a gun-wielding baddie while not firing, dodge bullets and kick him, and then, as he jumps away to another building, fire to hit his ankle as he lands (Ghost in the Shell)? How cool is it to see a half-goblin run though an encampment, dodging guards, arrows, and spears, dive through a wall (break it, not by turning ethereal), grab a previous gem, jump right back through the roof to avoid more guards, speed away, then turn at the last second to see an arrow strike his chest, and be pinned against a tree (Inu Yasha)? Who wouldn't be curious what will happen next when: a young boy wakes a demon (the hot chic variety), she is angry about the 700 years she spent imprisoned and playfully (well, maybe...) attacks him, destroys his school, then accidentally gets her hand cut off, leaves ... and when the kid returns home, he finds her sleeping in his bed (Tenchi Muyo)?
BTW, anime is great. And now that I think of it, here's a kickass webcomic. And here's a computer nerd style one.