Why did Japan fight China?

The answer above is technically correct, but obviously from the Chinese perspective. The rape of Nanking is very well documented and you can read it online. It also covers the history of this event.
Imperial japan was about expansion and military dominance. The Japanese people at the time were emerging from a period of rapid industrialisation. You have to remember that Japan only opened its ports to outsiders in 1853. Japan then had to modernise its army, move away from the Samurai rule etc.

At the time the Emperor of Japan was seen as a living god and he pushed for the glory of Japan and military aggression. The Japanese army were TAUGHT to believe that the Chinese people were sub-human (in fact all races were less than Japanese, but the Chinese were the lowest of the low). When the Japanese invaded China they met a token resistance and easily defeated it.

They set up their military command in old Nanking (or Nanjing). From there you can see the result of a conquering army de-humanising the local people. Rapes were a daily occurrence and many were beheaded and left to rot in the streets. The atrocities committed there are yet to be fully acknowledged and atoned for. This period is not taught in Japanese schools just as the crimes against humanity committed by the UK in many countries during the empire or the US against the native American Indians is not taught in the respective countries.

Half of this is mass brainwashing and psychology as well as land and resource grabbing and I can't fully answer it here.