Anime is a culture from Japan based on a style of animation which is now popular around the world.
The big eyes iconic of these illustrations originally comes from shojo manga, which focused almost exclusively on emotion. The bigger the eyes, the easier it was to communicate.
In time, the big eyes have become a trademark. This has been developed into many pop culture innovations pertaining to anime culture, such as the cartoon contact lenses featured below.
Another way cosplay fans (people who dress up like manga characters) can look the part is by using clever makeup. I think these look really cool.