They are built to conform to government regulations that tax small cars at a lower rate. This policy is designed to save space in crowded urban areas. It's also cheaper to insure and register a Kei Car.
Japanese drivers have to prove they own (or have leased) a parking spot before a car can be registered. This requirement is dropped for Kei Cars (in some areas).
Although kei car regulations are designed to combat urban crowding — most people who buy kei cars live in the countryside. They are far less common in urban centers. Tokyo people have money and aren't tempted by the lower tax rates.
Some people go so far as to buy large SUVs that are nearly impossible to drive and park in Tokyo's narrow, crowded streets.
Kei cars resemble toys. Kei car owners drive them as such. Kei car drivers are generally wild and dangerous.
Kei cars are designed for the Japanese market. They are exported to certain countries where there's a demand (mostly countries where it's difficult to find a parking spot).
The old standard for kei cars (before 1990) was 550 cc.
Before 1976, the kei car standard was only 360 cc.