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.