打 shows a hand 扌 driving a nail 丁 — the original sense was simply to strike. From hammering, it spread to become Chinese's do-everything verb: you 'strike' a phone call, a taxi, a game of ball.
打 is the Swiss-army verb of Mandarin: 打电话 (make a call), 打车 (hail a taxi), 打球 (play ball), 打工 (work a job), 打开 (open — with 开 you learned). Linguists count dozens of distinct uses. Master 打 and you gain fifty verbs for the price of one — just don't take 'hit' literally when a friend says 打个电话.
A hand 扌 hammering a nail 丁 — to HIT, and from there, to DO.
打的 (dǎdī, 'hit the taxi') entered Mandarin from Cantonese in the 1980s and is still the street word for cabbing.