Please note that kana is not a true syllabic script anymore. The reason for this is due to /n/. For example, take the word /sinbun/ "newspaper". If you break it into its syllables, it is sin.bun. While accents are determined by syllables in some dialects, kana--as well as Japanese speakers--segment this as si.n.bu.n. The appropriate term for this mora.
How did this idea to make a syllabary writing system come about in Japanese/Chinese history?
Kana are simplified forms of Chinese characters. Before the development of kana, Japan borrowed Chinese characters to phonetically write Japanese words. This is known as man'yoogana. Chinese of course is a different language with a different phonology, so there was considerable differences in pronunciation of the same characters. After doing this long enough, a short hand form of the same characters developed into the two forms of kana: hiragana and katakana.
Why did someone want to do it?
Until man'yoogana came around, Japan did not have a way to write their language. Hopefully we can agree that being able to write and record words is a useful thing.
Was the idea taken over from people who had already done it?
Quite likely. Records indicate that it was the Koreans who initially taught Japan about Chinese characters. Recent research has also indicated that katakana has its origins in the Korean peninsula (more precisely Silla). If you're interested in knowing more, search for the work done by 小林芳規, in particular regarding the text 大法広仏華厳経.
Also, where to the syllabary sounds come from?
The phonology of the language. Initially, the sounds were of the form ((C)V)+, so quite simply characters were applied to each of these fully formed sounds. A little later long vowels, palatalized consonants, and /n/ developed resulting in several innovations: doubling of vowels, small kana, and a new /n/. As the language continued to develop, several sounds naturally fell out of use and are no longer used.
For example, why is "yo" a syllable in the system but "ye" or "yi" isn't?
It may help to brush up on phonetics, particularly glides. Glides form a pair with vowels: the glide is non-syllabic while the vowel is syllabic. These sounds are phonetically quite similar and the lack of distinguishing is not an odd thing in human speech. You could try the linguistics forum for more details.
That said, /ye/ existed and was distinguished from /e/ in man'yoogana, the pre-cursor to kana. The distinction was lost by the time that the kana scripts developed, so no need for a character.
There is morphological evidence of /yi/ and /wu/. For example, /yi/ can be seen in the conjugation of the following verbs:
- kuy- (悔ゆ to regret): kuy-i, kuy-i, kuy-u, kuy-uru, kuy-ure, kuy-i
- mukuy- (報ゆ to repay): mukuy-i, mukuy-i, mukuy-u, mukuy-uru, mukuy-ure, mukuy-i
- oy- (老ゆ to age): oy-i, oy-i, oy-u, oy-uru, oy-ure, oy-i
And /wu/ can be seen in the following verbs:
- suw- (据う plant): suw-e, suw-e, suw-u, suw-uru, suw-ure, suw-e(yo)
- uw- (植う plant): uw-e, uw-e, uw-u, uw-uru, uw-ure, uw-e(yo)
- uw- (飢う starve): uw-e, uw-e, uw-u, uw-uru, uw-ure, uw-e(yo)
While they are apparent in the morphology, there is no evidence that they ever existed phonologically. You can handle this as a natural result of phonetics, or you could posit phonological rules such as /y/ --> ∅ / _ [i] and /w/ --> ∅ / _ [ɯ].