着 split off from 著 ('to place, notable'), keeping the senses of contact and continuation. What touches, stays; what stays, continues — the grammar of duration grew out of the physics of sticking.
As weightless zhe, 着 keeps actions alive: 门开着 (the door stands open), 听着 (keep listening), 拿着 (holding). As zháo it means managing to touch: 睡着 (fall asleep), 着火 (catch fire), 着急 (anxious — fire caught in the heart). Learners meet 着 everywhere and pronounce it wrong for months — the rule of thumb: after a verb describing a state, it's zhe; when something 'catches', it's zháo.
What touches, sticks; what sticks, CONTINUES — the -ing of Chinese.
睡着 (shuìzháo, fall asleep) vs 睡觉 (shuìjiào, to sleep): one character apart, and mixing them up is a rite of passage.