Here is a note from the 古漢語常用字字典 concerning synonyms for ‘quick, fast’ in Classical Chinese:
[辨] 快, 速, 疾, 捷. 這幾個字都有快速的意思.
“快” 表示快速是後起意義, 在上古只做愉快講, 而”快速”這個意義卻常用"速"表示.
"疾” 一般比”速”快一些. “捷” 指動作輕快, 敏捷.
(See under entry for 速)
[Note: I misread this dictionary entry initially - it doesn't mean that 'kuai' occured in a compound with 'su' in CC. The comments that follow have been edited in this light.]
This is saying what others have posted - that originally (or in the pre-classical period) 快 meant happy, and only later took on the meaning of fast. It was 速 which did mean ‘fast’ at that early date, which makes sense given the ‘movement’ radical. I would suggest that the semantic range of 快 may have expanded due to use as an attributive. For example, speed was a desirable quality in horses, so 快馬 was a horse that made people happy. Perhaps 速 lost its adjectival role and other words were used to fill in. An analogous usage in English would be “a goodly pace.”
You could see how this would work with 疾 as well. This character in Classical Chinese meant to be sick – in comparison to 病, 疾 denoted a less serious illness. If it was used to modify other words in the sense of “a feverish pace” or very fast, a similar semantic shift seems plausible.
I don’t think this interpretation necessarily contradicts @Michaelyus’s idea that this is due to 假借borrowing. This kind of semantic shift is known from many languages, as the English examples I’ve given suggest.
喜也從心夬聲
– Mou某♦ Sep 8 '14 at 4:50快,喜也。从心,夬聲
is the original text in Shuowen, so the core is why it can evolve to have a meaning "quick". I think there may be two possibilities: 1) some other character with a similar pronunciation had a meaning related to "quick", 快 was mistakenly written for that character, and then became popular; 2) 《說文新證》:“【夬】射箭時套在指上的扳指,分決為引申義。甲骨文从又,○形象扳指。" So if 夬 had a meaning related to a shooting arrow, it would be a reason for why 快 could have the meaning "quick". – Stan Sep 9 '14 at 18:37