How did the Japanese come to have three syllabaries?

Kanji appeared first thanks to cultural exchanges between early Japan, Korea and China. We don't know exactly when Japan began to use kanji but the earliest texts written in Kanji archeologists have found are from the 5th century C.E. (for example an inscription on a sword). The way to write Japanese with kanji for their pronunciation only is called manyôgana and it was the first way to write Japanese language (the most noticeable example is the poem anthology called manyôshû). There was also kanbun (that replaced manyôgana completely when it comes to writing official documents), that was a "sort of" classical Chinese used by Japanese people, that associated reading and writing of the kanji and was used for all officials documents. That's supposed to be Classical Chinese, but it was "highly modified" ... That makes kanbun extremely confusing ... So I add an example of kanbun so you can see the difference with "actual" classical Chinese (I know it's a wikipedia page, but I didn't notice any mistake in the example given and I think it explains well) : here

But manyôgana was not the easiest way to write Japanese. You had to learn hundreds of kanji, and if you didn't know the kanji, you were screwed. Besides, it was difficult for people to write with kanji, since it required a long education. That's the reason why kana appeared (the 9th century is the traditionally accepted date of the invention of kana).

But hiragana and katakana had different use:

Katakana were 男手 (otokode, men's scripture), and were used by Buddhist monk as furigana for difficult kanji and to annotate the sutra.
Hiragana were 女手 (onnade, women's scripture) and were at first learnt only by women of the Court, who didn't learn kanji. That's why many books written by women during Heian era use very few kanji ( here's a link to the original text of the genji monogatari ). At the end of the Heian era, they were also used by men, and were associated with text expressing feelings, such as love letters for example.

That's why there are three syllabaries : in the beginning they had very different use and were "adapted" to those who had to use them.

edit: I corrected some mistakes, clarified a bit the first paragraph and added a bit more about kanbun.