Funimation took advantage of its dual position (doing dubbing & butchery for US television aimed at the saturday-morning-cartoon set like 4kids and Nelvana while also producing reasonable translations & subtitles for anime fans who could afford DVD boxed sets) to weather through the industry crash in 2006, and bought up the dregs of ADV and other early movers with huge back catalogues. For the most part, Funimation’s “good anime” in its back catalog from prior to 2006 is actually ADV’s back catalogue.
(Somehow, they didn’t get everything — Evangelion seems to be out of their grasp, but Evangelion is strange in a number of ways, and the way that licensing negotiations occurred is one of them: it’s actually owned by Gainax instead of a production committee, which basically never happens except at Kyoani.)
In addition to swallowing up IP, they also swallowed up talent. The industry crash is the point at which non-Sunrise/Bandai dubs became watchable, because the best people from Funimation and ADV ended up working together. (Compare the Funimation dub for Steins;Gate to the ADV dub for Evangelion or the Funimation dub for Dragon Ball Z. The quality difference is huge, particularly in how much effort went into producing an accurate natural-sounding translation that matches the cadence of the original.)
Because they have the largest back-catalog of existing licenses, the most existing industry connections of all distributors, the highest-quality dubbing apparatus, and an exclusive distribution license for the biggest franchises, they were able to leverage this position starting about ten years ago to grow into a serious juggernaut: using control of existing licenses to get new licenses, strengthening their industry connections, and most recently turning better streaming services like Crunchyroll into part of their vertical integration apparatus instead of competitors (by sublicensing their licenses — i.e., Crunchyroll will not compete against Funimation for new shows, because any license Funimation gets will be sublicensed to Crunchyroll for simulcast).
Since the Crunchyroll deal, this situation has become pretty good for customers: almost everything worth watching is on both Crunchyroll and Funimation (with the notable exception of those older shows that Funimation doesn’t own the license to but that still have foreign licenses — HiDive seems to be snapping those up), and Funimation even rotates their popular stuff through Netflix and Amazon (trying, somewhat unsuccessfully, to do with them what they successfully did with Crunchyroll & become a middle-man for sublicenses with companies that know how to produce solid streaming services). Casual fans can get good stuff from Funimation’s back-catalogue from services they already subscribe to, while more serious fans can get almost everything between Crunchyroll and HiDive. Simulcasts and simuldubs have become normal, which old folks like me would have never imagined possible in 2005 (when even fansubs took weeks to come out).