Several parts combine into one character.

着 split off from the older 著 along an evolutionary fork. The core image is "to attach something to a body" — to clothe oneself. From this concrete root the meaning radiated outward: "to arrive" (your body attaches to the destination), "to adhere / cling" (literal sticking), "to focus on / become attached to" (psychological clinging). Then the three regions diverged dramatically. Japanese cleaved 著 and 着 into two distinct characters with separated meanings: 著 = "authorship / to make manifest" while 着 = "to wear / arrive / attach." Mainland Chinese kept the spelling 着 but loaded it with multiple pronunciations and grammatical roles.

Korean reading "chak." Used across all three branches of meaning in compounds: 到着 (dochak, arrival), 着用 (chagyong, wearing), 着陸 (changnyuk, landing), 密着 (miljak, close adhesion), 愛着 (aechak, attachment / fondness), 執着 (jipchak, obsession). One character, three semantic clusters.

Mandarin is where 着 gets genuinely complex — three distinct pronunciations: zháo (2nd tone) — 着急 (zháojí, anxious), 着火 (zháohuǒ, catch fire); zhuó (2nd tone) — 穿着 (chuānzhuó, attire), 着陆 (zhuólù, to land); and the unstressed grammatical particle zhe — attached to verbs to mark continuous aspect: 看着 (kànzhe, looking), 等着 (děngzhe, waiting). The grammatical zhe is one of the most common morphemes in spoken Mandarin, with no etymological resemblance to the noun-meaning of 着 — purely a phonetic loan.

Japanese on-reading チャク (chaku) covers the formal arrival/wearing senses: 到着 (tōchaku, arrival), 着陸 (chakuriku, landing), 着用 (chakuyō, wearing), 装着 (sōchaku, fitting / equipping). The kun-readings split the meanings cleanly: きる (kiru, to wear) — 服を着る (fuku o kiru, to wear clothes); つく (tsuku, to arrive) — 駅に着く (eki ni tsuku, to arrive at the station); つける (tsukeru, to attach / put on) — 印をつける (shirushi o tsukeru, to put a mark on). 着る is paired with 脱ぐ (nugu, to take off clothing).

Memory aid: clothing attaching to a body — wearing, arriving, clinging, all from the same image of contact.