快 = 忄 (heart) + 夬 (decisive). Its first meaning was a 'happy / pleased' heart — a heart that feels good. From the lightness of a happy heart came 'quick / fast'. So speed and joy share one character.
快 means fast AND happy, both rooted in a light heart: 快乐 (happy, 'fast-joy') — as in 生日快乐 (happy birthday) and 新年快乐 (happy new year); and 很快 (very fast), 快来 (come quick!). 快 also marks 'soon/about to': 快到了 (almost there). When you wish someone 快乐, you're wishing their heart to feel light and quick.
A light, decisive heart (忄) — QUICK, and happy.
快乐 ('happy') literally means 'fast-joy' — a light, quick heart. That's why every Chinese well-wish ends in 快乐: 生日快乐, 新年快乐.