Somebody asked me this a while ago and I couldn’t give them an answer at the time. I’ve spent a fair bit of time thinking about it.
First, best to clarify a couple of things.
Anime is not a genre. It is a medium. Specifically, it is broadly defined as animation created in (or under close direction from) Japan, targeted at a Japanese audience. The word “anime”, as used by the Japanese, means what an English speaker would call “animation.”
Secondly, only a fool would claim that all anime is good. As with almost any genre, it follows Sturgeon’s law: 90% of everything is crap.
To summarise; the main strength of anime is that it allows a very broad variety of styles and genres, with very little restriction. The better series use a solid plot to tie together interesting characters in a novel setting. Something like Beyond the Boundary, for example, has moments both tragic and humourous. The characters develop clearly through the series and have their lighter moments, but the two main protagonists both very dark backgrounds and can hurt others entirely by accident very easily.
By comparison, something like Neon Genesis Evangelion starts as a fairly straightforward giant robot show before ending in a fairly disturbing exploration of the human psyche.
Of course it also has its “cheesecake” series that approach soft porn at times; and a genre of “cute girls doing cute things” shows that seem quite pointless in many cases. However, the flexibility of the medium means that a show is not necessarily nailed down to a single idea or trope. Sturgeon’s law again, although the opinion of where the magic remaining 10% varies considerably between fans.
What almost all anime has in common is a continuity from episode to episode that lends the chance for buildup, rather than each episode standing on its own. It shares this with the best of television drama, while allowing more creative freedom.
Where it loses out in comparison with traditional non-animated drama is that it’s much harder to communicate subtle nuance of expression with animated figures. This is balanced a little by the existence of certain types of visual shorthand for the most common expressions. Unfortunately newcomers to the genre will typically miss these and so miss out on some of the depth of the shows.