When the Japanese walk into their house, one of the first things they do is take off their shoes. They leave their shoes in a vestibule called the genkan, and most people use special slippers to wear inside the house. (Sometimes there is a geta-bako, a shoe cupboard to store shoes.) However, they do not wear slippers in rooms floored with tatami because the scuffing can damage the floor, and separate slippers are used in bathrooms.
The genkan is an important place; it is the boundary between the outside world and the private world, the image of the home to visitors. There are even rules, or more like societal standards, for the genkan: You don’t step in the genkan barefoot. When visiting someone else’s home, you turn around upon entry and place your shoes facing outwards, so that they are easy to slip on when you leave.
This custom of removing shoes before entry originated as far back as the Heian period, for multiple reasons. With a wet climate, footwear was easily dirtied or muddied, and thus removing one’s shoes maintained cleanliness in the house. Furthermore, floor cleanliness was made especially important by the custom of sitting or sleeping directly on the floor or on cushions on the floor. Presently, although Japan has Westernised, its climate is humid, and it’s simply more comfortable to remove one’s shoes, or to wear sandals or slippers, because Western shoes do not provide sufficient circulation for comfort in the humidity.
Anime seems to capture the aspect… sometimes. It’s not like they’re going to stop and emphasise the removing of the shoes every time someone walks into a house; that would slow down the story progression unnecessarily. So they don’t often show the actual action of removing the shoes, but generally when characters’ feet are visible indoors, they have no shoes, although they generally just have socks on, not slippers.
Luckily, I did manage to find a precise example of this, Amano Yukiteru (recall that in Japan (or most Asian countries/cultures) the name is organised as surname, first name) removing his boots in the genkan. He doesn’t put on slippers though, just stays in his socks. This is from the anime Mirai Nikki (or in English, Future Diary). Good job, anime!