Anime tells stories in a visual format that for years and years you simply could not get anywhere else. Adult genre fiction was not well represented in American media. Even today, adult American cartoons are almost entirely "sitcom" format. They don't want to ever end, and they do not want to tell a story, they want to make more episodes.
Today, people are spoiled with their options for genre fiction. Cowboy Bebop blew my mind as a kid. Berserk was also one of my favorites. I did not think anything could be cooler than Vampire Hunter D. This was before Game of Thrones or The Expanse or any number of other amazing science-fiction or fantasy media we have today. Coming from a small town, it felt like a secret that nobody else knew about and I loved immersing myself in this grim, beautiful, foreign worlds[0].
A lot of anime has a lot of gross "fan service" that objectifies their female protagonists, and that is something I was never a fan of. It was something you tolerated because there wasn't anywhere else to go.
Lain is a great example of all of this. Where else are you going to get a gorgeously rendered cyberpunk-horror series?
0: Probably at a younger age than I should have been allowed to. Though honestly I don't see myself censoring everything from my kids. Part of the draw was the adult nature and feeling like I was getting away with something.