Why do characters in Japanese anime have ridiculously BIG EYES ?   

Well I don't really know the exact reason for this, in fact I too am puzzled to this feature of Asian anime, well Japanese in particular..
My 'theory' is, that Japanese animators tend to create these images to display what they actually want (rounder eyes) - maybe just a littele too exagerrated.. Right???
This 'wanting' for wider eyes came when the world started to get all Westernized.. Before, this never was an issue..
In fact, according a book written by Sonia Zaide eastern people think Western people are ugly, and this is the reason why the Easterns called the Westerns the 'The White Demons..' But as time went, the newer generation of Easterns thought of it the other way around especially when the Westerns are already winning the wars.. Inferiority came, that is...
It is also true that there are clinics in Asia that earn lots for making eyed round..
Well, anyway, as far as the cartoons are concerned, I dont see any major problem, I actually think it's cute..
Just hope I wont see anybody with such enormous eye size in real life...LOL
B-)

It actually has more to do with the history of Western cartoons than Japanese culture. Tezuka Osama, the creator of Astro Boy and Kimba the White Lion and a man commonly known as the "father of manga," created characters modeled after Western cartoons like Betty Boop, and he is credited with inventing the "large eyed" style that has become popular in modern Japanese anime and manga, particularly in "shojo" manga (girls' comics) Anime and manga featuring characters with much more normal eyes -do- exist, but they may be hard to find outside of Japan. I believe that Masakazu Katsura's I"s may be one series that actually does feature normal-eyed characters, but it's been a while since I've seen the interior art so I can't say for sure.