I think the answer is that it all happens the other way round.
Words are said, not read. The word for "today" is kyou. The way it's written is 今日, which looks like a fairly obvious way to represent the meaning. When you come to read it, you just have to know that it's kyou and not, as you might have expected, imabi or konnichi.
I guess that, since kanji were originally imported from China, 今日 is how "today" is (or was) written in Chinese. So the Japanese used 今日 to write "today" as well.