The word 'born' is the past participle of the verb 'bear' in its sense of 'to give birth,' descended from Old English 'boren,' the past participle of 'beran' (to carry, to support, to give birth). The Proto-Germanic ancestor was '*beraną,' and the PIE root is *bʰer-, one of the most fundamental verbs in the Indo-European family, meaning simply 'to carry.'
The connection between carrying and giving birth is a natural metaphorical extension that occurred independently in multiple branches of Indo-European. A pregnant woman carries a child; at birth, she has borne that child. Latin 'ferre' (to carry, from the same PIE root) similarly extended to 'fertile' (able to bear) and 'fetus' (that which has been born or is about to be). Greek 'pherein' (to carry) belongs to the same family
The modern spelling distinction between 'born' and 'borne' is a relatively recent convention, established by grammarians and dictionary makers in the eighteenth century. In Middle English and Early Modern English, the past participle was simply 'borne' (or 'born') in all senses, whether the meaning was 'carried' or 'brought into life.' The 1755 edition of Samuel Johnson's Dictionary began to codify the distinction, and by the nineteenth century, the rule was firmly established: 'born' is used for the birth sense ('she was born in 1990'), while 'borne' is used for the carrying sense ('he has borne a heavy load') and also for the birth sense when followed by 'by' ('a child borne by her').
Old English 'beran' was a strong verb (class IV), conjugating as 'beran — bær — bǣron — boren.' The modern verb 'bear — bore — borne/born' preserves this ancient strong conjugation pattern, making 'bear' one of the surviving strong verbs in English. The related noun 'birth' comes from Old Norse 'byrðr,' from the same Proto-Germanic root, brought into English by Scandinavian settlers in the Danelaw.
German 'geboren' (born) preserves the prefix 'ge-' that once marked the past participle in all Germanic languages. English lost this prefix during the Middle English period — Old English had 'geboren' just like German, but the 'ge-' eroded away, leaving only 'born.' The German expression for one's maiden name, 'geborene' (literally 'born as'), uses the same word.
The adjective 'born' meaning 'having a natural talent' ('a born leader,' 'a born musician') is first attested in the early seventeenth century, extending the birth metaphor to suggest that a quality was present from the moment of entry into the world. The phrase 'born again,' in its religious sense of spiritual rebirth, dates from the sixteenth century, translating the Greek 'γεννηθῆναι ἄνωθεν' (gennēthēnai anōthen) from the Gospel of John.