These three systems basically formed a Voltron of language if you will. Hiragana is commonly used to make up most of the sentence structure, katakana is commonly used for foreign words, and Kanji is most commonly used for nouns.
With their powers combined they make up sentences. As for the why, it largely has to do with tradition and history. Korean has shifted to a more easy to read/understand writing system, but in Japan it is much harder to do something like read a newspaper. Why? Because you either know the Kanji or you don’t. There is no sounding it out.
There are lots of exceptions to when to use certain writing systems. For example, sushi can be written like 送信, すし, or スシ in advertisements. I asked a Japanese friend about it and he basically said it just depends.