What the verb does now
In ordinary English, to leave is to go away. You leave a building, leave a job, leave the country. The same verb still does older work that has nothing to do with walking out: you leave a note, leave soup in the pan, leave money to a child. Those uses are not later extras. They sit closer to the oldest English sense than the everyday idea of departure does.
English also has a noun leave, permission or authorized absence, as in sick leave or to take leave. That noun is a homophone with a separate ancestry. The history below is the verb of remaining and departing.
The origin of the word leave is a story about causing something to stay, not about opening a door.
Inherited, not borrowed
The verb is native Germanic. Old English lǣfan meant to let remain, to let survive, to bequeath. A dying householder could lǣfan land or a spouse. Nothing in that core requires travel. Motion away is a later reading of the same act: if you make things stay, you are no longer with them.
The Old English form continues Proto-West Germanic *laibijan and Proto-Germanic *laibijaną, a causative of *lībaną, remain. To leave something was to cause it to remain. Close kin include Old Frisian lēva, Old High German leiben, and Old Norse leifa, still reflected in Icelandic leifa for food left uneaten. The intransitive remain-verb survives in Old English belīfan, Gothic bileiban, and German bleiben.
Written English has the verb from the Old English period, well before the Norman influx that brought so many French replacements. It did not need to be borrowed.
A causative on a sticky root
The deeper reconstruction is Indo-European *leip- or *leyp-, stick, adhere. Baltic and Slavic verbs of clinging (Lithuanian lipti, Old Church Slavonic lipeti), Greek lipos for grease, and Sanskrit lip- for smear or adhere keep that physical sense. In Germanic the cling became remain, continue: what adheres, stays put.
That remain-sense also feeds English life and live, historically about going on in existence. Older English lave for a remainder belongs to the same cluster. The morphology is real, not a folk tale: a causative suffix on a remain-root, unrelated to the plant word leaf.
A rival account once tied the Germanic family to Indo-European *leikw-, leave behind, the source of Greek leipein and Latin linquere, and so of relic, relinquish, and delinquent. The meanings line up neatly. The sounds do not, and that phonological mismatch is why the sticky-root explanation is the one usually kept.
From remain to depart
Around 1200, Middle English leven and related spellings take on go away, depart from, leave behind. Idioms of the period still show the hinge: leave the earth for die, leave the field for retreat. You go, and what was with you is caused to remain.
In the same stretch the verb covers stop, give up, omit, abandon, and allow someone to go. Leave out in the sense omit is late fifteenth century. Leave alone is about 1400; leave off as cease is from the same time. A spoken use of leave for let or allow, as in leave me go, appears by about 1840 and is mainly American; careful prose still prefers let.
The past form left looks irregular beside the infinitive. It is a weak past that changed in sound. Old English already used lǣfde and the participle gelǣfed. In Middle English the long vowel shortened before the following consonants, and d became t after f, giving left. The infinitive kept a long vowel and became leave. An older strong participle lifen is reported for the wider family; by attested Old English, lǣfan is already a weak class-I verb.
English once also used the verb intransitively: to remain, to stay behind. That sense died. The language kept the causative and the later depart readings.
The other leave, and a fossil in the past tense
The noun leave meaning permission is Old English lēaf, often in the form lēafe. It comes from Proto-Germanic *laubō and from *leubh-, care, desire, love. The first idea is approval that grows out of pleasure. That is why the noun stands near lief, love, and belief, and why German Urlaub and Dutch verlof mean time away from duty. The military sense of leave of absence is recorded from 1771.
A third verb leave, to put forth foliage, is simply leaf used as a verb. None of these explains left as the opposite of right, which is a different adjective, probably from a sense weak.
The compact takeaway is the past tense. Left is lǣfde after shortening and unvoicing, a dental suffix on a causative that meant make remain. When someone left the keys on the table, English is still, under the sound change, saying that the keys were caused to stay.