I hope this is OK, I'm not a professional historian, but I was a karate-ka for about 20 years. And part of that was reading. So I hope I can provide an in depth answer that satisfies requirements. A lot of my reading comes from Randall Hassell's "Shotokan Karate: Its History and Evolution" and Gichin Funakoshi's "Karate-do Nyumon," and his "Karate-do Kyohan," as well as what I can recall from his autobiography "My Way of Life." I don't have the latter in front of me, so it will rely on memory a bit, sorry.
Now, this is told from the perspective a Shotokan Karate, which I understand to be the origin of the name. The beginnings were in Japan, in the Ryukyu islands, specifically Okinawa. The stories of Funakoshi learning Karate from Itosu and Azato, who are part of the history of a fighting style without traditional weapons because of the banning of weapons during the Meijii Restoration...I'm not competent to assess. Hassell says that there are Chinese roots as well for this style.
In 1924 Funakoshi was asked to teach a small group of Japanese university students (in addition to the private lessons he had been teaching prior. College clubs began coming together around this time. this also led to some splintering of Shotokan, but apparently without rancor.
Masatoshi Nakayama described training with Funakoshi as involving Kata (forms), followed by Makiwara practice (a board wrapped in rice rope and setup to practice striking). Hassell quotes him saying "When he [Funakoshi] selected a kata for us to practice, we would repeat it 50 or 6t0 times, and this was always followed by intense practice on the makiwara. We would punch the makiwara until our knuckles were bloody....the training was so grueling that of the 60 or so freshmen who enrolled with me in 1932, only six or seven of us made it through the first six months of training." He notes the around this time Funakoshi began teaching gohon kumite, a type of ritualized sparring practice drawn from kata. This evolved into kihon-ippon kumite and then into the free sparring techniques over a period of five years. These ideas are through out Kyohan.
And here is the key crossover moment:
in 1953, the U.S. government prevailed upon Funakoshi to demonstrate karate for members of the Strategic Air Command (SAC) at several Far Eastern bases.[From later in the Hassell's book, I think this means Eastern US, I'm not positive on this.] At the age of 83 , this remarkable man boarded a U.S. government plane and, accompanied by Masatoshi Nakayama of Takushoku University, Toshio Kamata of Waseda University, and Isao Obata of Keio University, toured many U.S. air bases and demonstrated karate for thousands of members of SAC.
If we fast-forward past Funakoshi's death in 1957, we see the development of the Nihon Karate Kyokai (Japanese Karate Association, JKA) as a group that saw themselves as the overseers of Karate in Japan. The JKA began a somewhat divisive process of creating sport Karate. Nakayama talks about the dangers inherent in this creation, saying "My greatest concern at that time was to ensure that karate, if given a sporting aspect, would not lose it's essence as an art." It is around this time (mid 1950's) that the JKA develops it's Instructor Training Program (which, I'm relatively sure continues today, some friends of mine have gone through it in the past few years).
Back to the demonstration at air bases. According to Hassell, the "American interest was do great that many airmen began seeking instruction, and soon Karate and Judo clubs were established on the bases." Nakayama, apparently a big cheese in these demonstrations, begins to see difficulties in education Americans, in large part due to the huge difference in cultures (if you're interested in this, I can't recommend "Chrysanthemum and the Sword" enough.) The question of "Why?" Hassell notes as a new one for Nakayama.
The first official JKA instructor in the US was Teruyuki Okazaki who started in Philadelphia in 1961. I'm lucky enough to know a few people who have trained with him and say that he was a remarkable man. The JKA began starting dojo's in cities across the US at this time. Kansas City, New Orleans, Los Angeles, Denver, Hawaii, Chicago, Florida, New Jersey, Arizona. An organization started by Hidetaka Nishiyama, the All American Karate Federation, applied to the AAU in 1970, listing more than 20,000 members at the time. And I think there is your flood gate.
I hope I did justice to the material. Hassell's work is great, but I don't know if you can find it for a reasonable price (I think it is out of print. My copy, a second edition (ISBN 0-911921-09-5) is spiral bound.