For a long time (starting in the 1960s and ending sometime in the 1990s), foreign distribution licenses for anime were so inexpensive and so unrestrictive that it was common practice for distribution companies to buy the distribution rights to an anime & treat the frames as raw material for creating essentially completely new shows (the same way that Adult Swim treated Hanna Barbera cartoons). As a result, a lot of children’s programming during the 60s, 70s, and 80s outside of japan was actually repurposed anime (for instance, Speed Racer is an anime & certain plot elements were changed or introduced with the dub, Robotech and Voltron are both collections of episodes of anime from different franchises stuck together and recut with new stories, and the original Gachhaman anime has been adapted for american audiences no less than six times under six different names with different plots). To some extent, this continues: distribution companies that focus on children’s animation, like 4Kids and Nelvana, buy up cheap franchises and heavily censor/localize them as saturday morning cartoons.
By the late 1980s, there was a growing contingent of non-japanese folks who recognized that the shows they grew up watching were anime, and they began to hook up with the thriving japanese tape-trading network, which had a heavy overlap with anime otaku-dom (tape traders were fairly frequently also owners of model kits & garage kits, and subscribers and contributors to the growing anime zine culture). This hookup is the ancestor of fansubbing: individuals that were part of tape trading networks would smuggle tapes overseas, where they would be subtitled & distributed to the small but growing underground of foreign anime fans. (Notably, some members of this cohort were among the production design crew on Star Trek TNG: eagle-eyed fans will find a bunch of references to the popular anime Dirty Pair on props and in the backgrounds of shots.)
By the early 1990s, the anime underground in the united states had begun to spawn conventions, at which point distributors like ADV, US Manga Corps, and Central Park Media appeared. These new distributors chose licenses to appeal to this otaku audience, who were adults & also hipsters. As a result, they focused on licensing OVAs (which tend to be shorter than TV anime, and also lack content restrictions: OVAs in the 1980s and 1990s were famously gory, violent, and full of sex — and often in fact full of gory sexual violence). Licensing short OVAs meant that a franchise didn’t have many episodes, so you were locked into producing fewer tapes: if a three-episode show like Cyber City Oedo 808 were to fail, you’d only have two or three underperforming tapes to keep in stock, while a 26-episode series like Evangelion, if it didn’t sell, would have up to 13 underperforming tapes that needed to be kept on shelves or made available for back-order. Additionally, translating & dubbing a shorter show was less expensive. Today’s anime fandom outside of japan really comes from the community these folks were marketing to, culturally, and older fans (like myself) will have been a part of this community, which didn’t really die out until the crash in 2006 (which bankrupted most of the big players in american anime & manga licenses, including ADV and Tokyopop).
In the late 90s, Cartoon Network launched Toonami, which was an attempt to link together the saturday morning cartoon crowd (already watching shonen staples) and the OVA and tape trading crowd (who had already seen the edgier OVAs for some of the series being broadcast). Then, in 2001, they launched Adult Swim, which was aimed squarely at this older OVA & tape trading crowd. A lot of teenagers got their first taste of anime aimed at older audiences through this block, & its success led to renewed interest among early adopters (like TechTV & the SciFi channel, both of which had done earlier experiments in broadcasting anime aimed at adults). Adult Swim had a huge impact on how anime fandom in the US saw itself & on what franchises became popular. The new fandom swarmed to the existing distributors and to fansubbers, who had by this point moved away from tapes and toward distributing bootlegs over the internet.
For fansubbers & the communities around fansubbed show (which were still the majority of shows — until the current state of constant scramble over simulcast licenses between Crunchyroll, Funimation, Amazon, Netflix, and HiDive was established around 2010, the only shows that got licensed were the ones that somebody thought could get shown on Adult Swim, marketed to the Adult Swim viewership, or chopped to pieces for saturday morning cartoons), the difficulty of access was part of the appeal: anime fandom was elitist and exclusive, and newcomers were expected to learn how to use IRC to download episodes, learn a bit of basic japanese to understand honorifics, and download special codec packs and special video playing software. Specialized discussion forums like 4chan popped up.
In 2006, Crunchyroll appeared. It used youtube-like video streaming tech to stream a collection of fansubs the creators had downloaded (a lot like KissAnime today). As far as I know, it was the first anime streaming service. After the crash, Funimation (formerly a 4Kids/Nelvana style saturday-morning-cartoons chop-shop) bought up licenses and staff from the now-floundering major distributors, and made a streaming service as an alternative way to access their now-substantial catalogue of licenses. Crunchyroll went legit, buying licenses for all sorts of shows. Anime streaming undercut the bloated pricing of anime DVDs (which used to go for $40 a DVD, with three episodes to a disk, for popular or long-running shows back in the bad old days, when ADV would charge you a hundred bucks for an empty box to put your DVDs in). The competiton between Funimation and Crunchyroll, along with the apparent massive market growth (which was really a side effect of people ditching their habit of buying overpriced DVDs in favor of watching a lot more anime at a much lower price on streaming sites) led to more companies getting in the game, including Netflix and Amazon, which already had big audiences. When they started getting anime licenses, they ended up exposing anime to even larger audiences than Adult Swim did.
That brings us to today: a bunch of distributors, importers, conventions, and community hubs that have one foot in a thirty-five year history of bootlegs, encouraged by steadily growing audiences to go legit and more or less succeeding, with each step toward legitimacy profiting off the previous, more hard-core audience while increasing access.
(This article adapted from an answer on quora)