Hand radical on the left (the side-form of 手) — the indexing radical, marking 持 as a hand-action. To 持 is literally 'to hold in the hand,' from which came 'to maintain, sustain, manage.' Family: 打 (hit), 抓 (grab), 拿 (take), 推 (push), 握 (grip).
Right 寺 supplies the sound — sì drifted to chí, a regular Old Chinese alternation. Same phonetic in 时 shí (time), 等 děng (wait), 特 tè (special), 诗 shī (poem). 寺 itself is 土 over 寸, originally 'office/court'; 'temple' is a later meaning.
