Why it was primarily in America, but not in those 2 regions?
Hello! As a game dev and general gaming history nerd I'd like to try my best and answer this question!
Firstly, let's talk about the name of the event, the Video Game Crash of 1983. It wasn't actually so much a crash of the video game industry as whole, but rather a crash of the American video game home console market, as during this period home computer games and arcade machines actually continued to be developed, sold and played without much hinder.
So with that out of the way, let's look at the markets in 1983 in America, Europe and Japan, and why the American one crashed and the others didn't.
THE CRASH IN AMERICA
In America the video game home console market had been booming and expanding rapidly since the beginning of the 80s. Atari was riding the waves of succes and numerous companies such as Magnavox, Mattel and Coleco were trying to follow suit. Stores were setting aside large amounts of store space and inventory to display all the various game consoles and their libraries. And these game libraries were getting gigantic, ever since Activsion had been able to successfully become a third party game developer in 1979 scores of game development companies rose up, trying to emulate their success. As the consoles of the time had little in the way of anti-piracy measures, everyone could develop and release a game that would run on them once they had the specific know-how, which spread quickly.
This all led to two problems, market saturation and overproduction. By 1983 the various video game home console manufacturers where fighting one another for dominance on the shelves, but they weren't just fighting amongst themselves. Third party developers released their, at times awfully low quality, games for lower prices than those of the console manufacturers and home computers were becoming cheaper and more powerful. A price war erupted not only between first and third party game developers, but also between video game consoles and home computers. Why buy a new video game console when you could buy a home computer for a similar price, which could play games with better graphics and mechanics and also do so much more?
Meanwhile Atari and the other home console manufacturers where still producing greater amounts of games and consoles, expecting the record growth to continue. But sales started to slow down and the market became over saturated with third party games, gigantic game libraries and numerous consoles. Stores didn't have enough retail space to show all of the consoles and games in large amounts, which led to lower game and console sales than expected by the companies. Lower sales led to less investment in new innovations and technologies, which led to stagnation as the market for the old consoles was saturated and consumers didn't see the need to buy new ones, as there weren't any flashy new features.
Atari, the king of the console market, had also bought the home console licenses for the blockbuster movie E.T. and arcade hit Pac-Man for a hefty sum, started gigantic advertisement campaigns and rushed the two games into production so that they could be released for the holiday seasons. The games were badly received, less than half of the millions of copies were ever sold. and Atari was stuck with 6 million unsold E.T. cartridges, 5 million unsold Pac-Man cartridges and numerous consoles, which they buried in Alamogordo, New Mexico. Atari didn't manage to get their return on the investment in the millions of cartridges and expensive advertisements and licenses.
A string of bankruptcies followed as stores began to return unsold inventory to the respective companies, who didn't have the money or goods to reimburse them. video game home console manufacturers went out of business, stores stopped carrying their products, believing that the video game craze was over.
ATARI SHOCK IN JAPAN
While the Japanese market itself wasn't very much hit by the crash, Japanese game companies were still surprised by it, and dubbed it "Atari Shock". Japan wasn't hit by the crash because the Japanese home console market wasn't intertwined with the American one at all, with Japan having its own homegrown companies and consoles. Japanese companies also continued to innovate in the home console market and in 1983, the year of the infamous crash, we saw the release of the Nintendo Famicom and the SEGA SG-1000, among others. Due to strict anti-piracy measures inside the consoles itself, it was also much harder for unlicensed third party developers to develop games for Japanese consoles, so the companies could very easily control the library of games for their system. Due to the Japanese market being almost completely independent of the American one, coupled with continued innovation, the Japanese market was able to avoid saturation and wasn't take down with the American market when it fell due to not having many ties to it.
WHAT ABOUT EUROPE
Europe also wasn't very much hit by the crash. In part this can be attributed to the fact that, like Japan, the European home console market was shared with native companies such as Phillips. But the home console market was also incredibly small when compared with the home computer market. To give an example, from 1980 to 1982 Atari only sold 30.000 Atari 2600 consoles in the Netherlands, which had 14 million inhabitants at the time. And not only were home computers more common, their games were also a lot cheaper, being on tapes or diskettes instead of expensive cartridges. Commodore had also launched an ad campaign to convince people that home computers were more educational then home consoles, which led to stores not carrying home consoles and software developers not developing for them. Thus when the crash happened, there wasn't actually anything to crash in Europe, the market simply didn't exist.
I already tackled this question from the perspective of the British market specifically, and largely what's said applies to all other territories. I will elaborate a bit on this though.
The biggest obstacle to there being a market to be disrupted (which is what a crash is) lays in the fact that there wasn't much of a console market in any other region to overturn. The console market operates on different economics from the computer and arcade markets which were active at the time.
In console manufacturing, a software creator (for the early period we're talking about here, 80% or more of the software was controlled by the hardware creators) would have to lock down a cartridge several months in advance, sometimes up to 6 months (such as in the case of Howard Scott Warshaw's E.T). This allowed time for the technological problems to be sorted out for production and allow for a lead time to make plenty of these specialized cartridges. Because cartridges take a long time to make, a manufacturer would have to estimate how much they were going to sell on the first go around and hope they sell them all at full retail price. If it was super duper popular, they may be able to do a second manufacturing run, but these economics were turbulent and the turn around was too long to be making major overestimations or underestimations.
Atari, for a huge myriad of reasons, overestimated how much the entire market could sustain. According to people who were there interviewed by my friend Alex Smith, they did not do something like make more Pac-Man cartridges than there were systems like the other commentor says, but across their wealth of games they made excess millions that once the bottom dropped out of the market they couldn't do anything with. The E.T. landfill was,
The other factor that plays into this are the third party manufacturers. Basically let's assume that the market can sustain 100 cartridges. Atari, not wanting to cede ground to a competitor, makes 100 cartridges. Activision and Imagic each make 20 cartridges. All the other competitors combined make 70 cartridges. That's 170% of market demand which, no matter how good or bad a game is, ultimately leads to the whole industry getting burned.
So now to the question of outside the US. In 1982, there were about 10 million Atari VCS systems on the market around the United States. There may not have been one million sold in Europe or Japan combined (though perhaps if you take every territory together it may amount to a million). While Atari did make an attempt to push their system in these territories, especially Japan and the UK, they really couldn't have broken through at that time.
The major issue was cost. Europe has always - in general - had less general spending money than the United States in it's economy. Therefore the most popular systems throughout history have been those which have gone down in price due to ease of manufacturing. In the pre-Atari VCS days, local made Pong clones were more popular than Atari's or Coleco's offerings in that category. In the NES-era, the Sega Master System was a cheaper alternative to Nintendo and so took up the market in most of Europe. The infamous price war of the Sony Playstation helped it gain traction in Europe as well. During the early 80s, there was no cheap alternative to the VCS. Therefore, people just didn't buy consoles.
In Japan it was a different matter. As has been true for most of the post-occuptation era, Japanese companies don't like dealing with foreign entities on their own soil. They'd rather a company abdicate almost full responsibility for a Japanese territory to a native Japanese company, therefore creating a middle-man affiliate. It's not that this doesn't happen in other countries, but it basically limits the amount of direct market control that a corporation can have over their product in that region.
For some reason, in Japan, the toy company Epoch decided to redesign the VCS to be styled as the Atari 2800 with a total cosmetic redesign. Maybe this would have been not such a big deal if not for their timing: May 1983. Two months later, the Nintendo Famicom was about to come in. By this time, Atari didn't have much money to support the foreign operation. As with many things, there aren't good numbers on how much the VCS or any other Japan-foreign console sold, but the penetration was nothing. Epoch moved on to the Super Cassette Vision rather quickly.
So in total, in all of these territories, the market wasn't there to disrupt. Yeah, sure, maybe some well-off children were sad that no new games were coming out for their 2600 or Colecovision, but in general they hadn't been buying that many cartridges anyways. Foreign games continued to make influences on both Japanese and European creators, but they didn't need to reach a wide audience to do that. Systems like the C64 would penetrate in a mass-market way to the European territories especially so the fast action games which could be played on a console could be experienced in a different format.
Perhaps the misunderstanding is in assuming that in the 1980s that companies were the same sort of global operation they are today. It was all very fragmented, and the money wasn't generally there for the console companies to care about how much their European divisions were really drawing in. Some, like Activision, made smart choices in investing in the European market but they never relied on it. In the United States, retailers had a problem. In the rest of the world, it was a gradual diminishing of a stock that was already in short supply.
Sources
Console Wars by Blake Harris. The Nintendo chapter of this book provides a good view of the retail economics in the United States and why it collapsed.
Atari Inc. Business is Fun by Marty Goldberg and Curt Vendel. The many myths and legends of Atari delved into, but it doesn't cover the business end nearly as well to explain the causes of the collapse.
Zap!: The Rise and Fall of Atari by Scott Cohen. Slightly better on examining the business side of Atari's demise, though it doesn't quite put all the pieces together.
Epoch and the Cassette Vision – 1997 Developer Interview translated by Alexander Highsmith. Gives an overview of the Japanese industry including Atari's attempts to enter Japan.
Replay: The History of Video Games by Tristan Donovan. As a more European-centric look at the industry, it briefly covers some of the dominant consoles in those regions.
Video Games Around the World edited by Mark J. P. Wolf. A highly uneven tomb that occasionally provides some fascinating insight into local industry histories. The Western European situation is the best explained of the lot.
They Create Worlds Podcast by Alex Smith and Jeffery Daum. The Crash is covered in many various ways throughout the entire podcast based on insider interviews and a very large section of general press materials. Proper citation will be in order when his book comes out in December, but he's already a verified Wikipedia-citable source through his blog Video Game Historian.