Anime popularity in western countries such as the United States owes a lot to the 1980s. Whereas American properties such as Star Wars and Blade Runner influenced plenty of anime of the time, it all found its way right back. There was already something of a market for Japanese imports in the US thanks to properties like Ultraman having found success on syndicated television a decade prior. The growing action figure market of the 80s – which was once again propelled by Star Wars a few years early – blew the doors wide open.
Thanks to a time where cartoons could be made based on toys, it became lucrative for American toy and media production companies to license materials already made and simply repackage them in product and on their website. Voltron and Robotech exist as notable examples, where existing Japanese properties like Dairugger, Dougram, Macross, and GoLion were cobbled together into new franchises and edited together for American tastes. Gatchaman was also brought over in the mid-80s as G-Force. The massive hit show The Transformers is also worth noting; while its cartoon was comprised of original material overseen by Hasbro and Marvel Comics, it was based entirely upon mecha toys initially designed and released in Japan.
So, the Japanese flair of anime was already infiltrating the western market in full swing by 1985 or so. From there, the advent of VHS and its corresponding rental chain stores slowly gave even more access to anime properties, and the entire genre began getting a word of mouth acceptance among circles invested in the likes of sci-fi, comic books, and animation in general. It wouldn’t be long before notable Hollywood figures like Steven Spielberg could be found praising the likes of Hayao Miyazaki’s Lupin III film, Castle of Cagliostro. Anime was very much on its way in, and it wasn’t long before its western popularity hit critical mass in the 90s.