are available. It used to be de rigeur for parents to buy a big set of 'World Literature' books (as well as a set of 'Japanese literature' books), rather like the way encyclopedia sets used to be bought by many American families. My parents certainly did, and expected the kids to read them all. In school kokugo (Japanese, or the native language) classes, we usually had at least 2-3 excerpts from translated works from the west to study every year. So there is a fundamental familiarity with translated works. I noted here: Makiko Itoh's answer to What is the public view of Germany in other countries today? the familiarity Japanese people have with German literature; this holds true for the literature from many other nations. It should not be surprising that the Anne series was chosen from Canada, since L.M. Montgomery is probably Canada's best known author elsewhere (maybe Margeret Atwood can claim that title now).
Anne of Green Gables was first translated into Japanese by a woman called Hanako Muraoka in 1952, with the title " Redhead Anne" (赤毛のアン, akage no an) It seems that she omitted several passages and so on, which doesn't make them the most accurate translations, but she certainly was a very skillful translator. The first Anne book was a huge hit, and the rest of the series was also translated. Since then the series has been translated several times by different people.
Here's the cover of probably the earliest version of the book. (It was an insert or 'furoku' for a popular girls' magazine.)
The typical Anne reader first encounters the books when they are tweens. Most if not all Anne fans are women. Japanese women tend to cling on to things they were fond of as children or teens well into their adulthood. For instance, adult women still love things like Hello Kitty, or wearing clothing with cute characters on them, and no one thinks they are being childish or whatever. (Some western observers take this clinging onto childhood things as a sign of the infantility of the society. Take that for what you will.) So once an Anne fan, always an Anne fan, for many people.
I think that the Anne stories hold so much appeal because of both the character of Anne, and the rural setting of Prince Edward Island. Anne is a cheerful, smart, hardworking girl who overcomes many hardships. The fact that she's an orphan is significant too. Orphans doing well type stories exist in many cultures, but they are particularly popular in Japan. Another feature of the Anne books is the country setting. Western country style decor, fashion and so on have been popular on and off for ages, since about the 1970s. The way Anne dresses, the things she cooks (and the books abound with such details) are highly appealing to some girls.
The Anne popularity was given a big boost in popularity for a new generation by the first anime series mentioned, which was part of the World Masterpiece Theater series (世界の名作劇場); it first aired in 1979. At one time the series was must-viewing in many Japanese households, and a lot of kids got their introduction to various European works that way .
Other books with similarities to the Anne series were popular too, such as Little Women (the Japanese title is 若草物語 wakakusa monogatari, which means "Young Grass Story" - no idea why they got to that title), Pollyanna and Pippi Longstockings. Little Women has also been made into anime a couple of times, though not as part of the World Masterpiece Theater series. Also, another teenaged girl who is (or was) popular in Japan is the real-life Anne Frank. Brave girl heroes abound in popular culture such as manga and anime, and in her way Anne is a brave girl hero.
I'm actually not sure if the tweens and teens of today still read books like the Anne series as avidly as girls from the 1950s-80s did. But they are still available, and pretty popular. Many adult women yearn to make a pilgrimage to PEI and the "original Anne house", and quite a few do. Several Japanese tour companies in Japan and the U.S./Canada have Anne-centric PEI package tours to accommodate them.
There's an amusement park called Canadian World Park in Hokkaido, which has the climate and landscape most similar to eastern Canada The star attraction is "Anne's House". (Note: I previously said it had gone out of business in 1997 but I was wrong. It went backrupt in 1997, and was taken over by the local municipality, Ashibetsu City. It's still open from late April until the 3rd week of October. Japanese site: http://www.city.ashibetsu
 The original question stated that it's been part of the school curriculum since 1952, but this couldn't have happened since the first translation appeared in that year. I looked up the contents of kokugo (Japanese) textbooks from the past from a couple of major textbook publishers, and found just one instance of an excerpt from Anne of Green Gables listed, for a first year of middle school (junior high) textbook from one publisher in 1975. Interestingly, the one overseas author that appears the most in middle school textbooks is Hermann Hesse, and second most popular seems to be Maupassant. Russian literature is also frequently featured. Incidentally, Chinese literature is studied more in a separate class called koten (classic literature), in kanji.
 By far the most popular World Masterpiece Theater series was the first one, Heidi, or "Heidi, Girl of the Alps" (アルプスの少女ハイジ, arupusu no shoujo haiji) based on the book by Johanna Spyri. Japanese people make pilgrimages to Heidi-land (the Grisons in Switzerland), just as they do to Prince Edward Island.
The second most popular series was probably A Dog of Flanders (フランダースの犬 furandaasu no inu), based on a fairly obscure novel by an English author called Ouida (Maria Louise Ramé). Japanese people make pilgrimages to locations in that one too, especially Antwerp Cathedral where Nello, the boy hero of the story, dies at the end.
In both cases the natives scratch their heads in puzzlement as to why Japanese people are so fond of these characters, especially for the Dog of Flanders. (At least in Switzerland Heidi is universally known, but A Dog Of Flanders is pretty obscure in Belgium. But both locations benefit from Japanese tourists.)
 Related: Makiko Itoh's answer to What are some of the greatest depictions of feminine bravery in the arts?