快 originally depicted a heart (心) with a stroke suggesting opening or relief — its earliest meaning was "pleased, delighted." By the Tang dynasty it had extended to the sense of "swift, fast," perhaps because a relieved heart feels light and moves quickly. Both senses ("happy" and "fast") survive in modern Chinese: 愉快 (yúkuài, happy) and 快速 (kuàisù, fast/rapid).

快 is extremely versatile in everyday Mandarin. 快来！(Kuài lái! — "Come quickly!") is heard constantly in homes and classrooms. It also appears in the near-future construction 快…了 (kuài … le), meaning "about to / almost": 快到了 = "We're almost there." Don't confuse 快 (kuài, fast) with 块 (kuài), the measure word for yuan or chunks — they are homophones but completely different characters.