Origins
The word 'page' — so fundamental to literacy that we barely notice it — conceals an agricultural metaphor stretching back to the Roman countryside. It derives from Latin 'pāgina,' which in its earliest attested sense meant a rectangular frame or trellis to which grapevines were trained. The columns of writing on a Roman papyrus scroll, arranged in neat vertical strips with regular margins, evidently reminded someone of vines fastened to a trellis, and 'pāgina' made the leap from vineyard to manuscript.
The Latin verb behind 'pāgina' is 'pangere,' meaning 'to fasten, to fix in place, to drive in (a stake),' and by extension 'to compose in writing' — that is, to fix words permanently. The past participle 'pāctum' gave English 'pact' and 'compact' (things 'fixed' or agreed upon). The connection between 'pangere' (to fasten) and 'pāgina' (a trellis, then a page) is the idea of orderly arrangement: vines are fastened in rows, and so are lines of text.
A related and fascinating branch of this root produced 'pāgus' — a rural district, the countryside, a village with its surrounding farmland. From 'pāgus' came 'pāgānus,' a country-dweller, a rustic — and eventually 'pagan,' because Christianity took hold in Roman cities before it reached the countryside, so 'pāgānus' (villager) came to mean 'non-Christian.' The vine-trellis, the page, and the pagan all share the same Latin root.
Latin Roots
In classical Latin, 'pāgina' referred specifically to a column of text on a scroll, not a leaf of a codex (bound book). Roman scrolls were written in columns roughly 5–8 centimeters wide, and each column was a 'pāgina.' When the codex — the ancestor of the modern book — replaced the scroll between the second and fourth centuries CE, 'pāgina' transferred from the column to the leaf, and eventually to one side of a leaf.
The word entered Old French as 'page' and from there into Middle English, where 'pagine' is attested from around 1440. The simplified form 'page' became standard in English by the sixteenth century. The English word 'page' meaning a young attendant or servant (as in a page at court) is an entirely different word with a different etymology — from Greek 'paidion' (child), via Italian 'paggio.'
The modern derivatives of 'page' include 'paginate' (to number the pages of a text, attested from the seventeenth century), 'pagination,' and the now-ubiquitous 'web page,' which extends the ancient metaphor into the digital realm. A web 'page' is not a physical leaf, of course, but the metaphor of a defined area of organized content — something to be read in a single view — carries over naturally. The trellis has become a screen, but the idea of orderly rows of content, fastened into a frame, persists.
Modern Legacy
'Propagate' is a cousin worth mentioning. It comes from Latin 'prōpāgāre,' to extend by means of 'prōpāgō' (a vine slip or cutting used for transplanting), from 'pro-' (forward) and the 'pag-' root (to fasten). To propagate was originally to fix new vine cuttings into the ground — to spread a vineyard. The agricultural imagery of the vine thus runs through an entire constellation of English words: page, pact, pagan, propagate. Each preserves a different facet of the Roman farmer's world, fixed permanently in language.