Ideogrammic compound (會意): 大(“man”) + 口(“mouth, object”). Top figure simplified to 土, unrelated to 士 (“scholar, gentleman”), while bottom simplified to 厶. There are various interpretations of the combination of “man” and “mouth, object”. One is that the ideograph represented a man with a hole marked in his crotch, and the inventors of writing had perhaps “anus” (i.e. getting rid of) in mind (Schuessler, 2007). Another is that it represented a man departing from a cave or city.
Alternatively, it may be the original character of 呿 (OC *kʰal, *kʰas, *kʰab, “to open one's mouth”), the meaning given by the combination of 大 (“big”) and 口 (“mouth”). The meaning “to depart” would be an extension of the meaning “to open one's mouth”, as the lips depart from each other when one's mouth is open.
Shuowen Jiezi interprets the character as a phono-semantic compound (形聲, OC *kʰaʔ, *kʰas): semantic 大(“man”) + phonetic 𠙴.
In addition, 去 is also the original character of 盍 (OC *ɡaːb, “to cover”). The 大 is a cover on top of an object 口.
There were two pronunciations in Middle Chinese. The rising tone pronunciation originally meant “to put away, to eliminate” (causative of “to go away”), and the falling tone “to go away, to leave, to depart” (anticausative). The merger of the two pronunciations already happened in Old Chinese and most extant dialects do not observe this distinction now.
This word is of Sino-Tibetan origin with the basic meaning of "to get rid of". Compare Tibetanསྐྱག(skyag, “to spend, to lay out; dung, excrement”), རྐྱག(rkyag, “dirt, excrement”), Burmeseကျ(kya., “to fall, to become low”), ချ(hkya., “to put down, to bring down, to lower”).