This is because during the Tang dynasty, when Chinese writing began to migrate into Japan on a large scale, China was the single most important cultural power in Eastern Asia. The Japanese did not borrow Chinese writing to write their own language, they borrowed the Chinese language as the language of education and culture. Early Japanese writings are essentially written in classical Chinese. To be able to write in medieval Japan basically meant to be able to read and write Chinese. This language was called by the Japanese kango, that is, Chinese language.
Later on, as the need to write extended to the general population, it became important to be able to record spoken Japanese language. This began with using the Chinese characters for the phonetic values and then by developing two kinds of unique phonetic systems. These two systems are called hiragana and katakana and are still used in conjunction with the Chinese characters. Thus Japanese people today write with a combination of three different kinds of scripts, which together are perfectly capable of recording their native language.
Although today, in the age of Pokemon, manga, and Playstation games, it is well-known that Japanese use Chinese characters, people are less conscious of the fact that other nations in Eastern Asia used to use Chinese characters too. The most important of these were the Koreans and the Vietnamese.