My guess is that it has to do with the physical process of scanning, moving your eyes over the words, but I can't find any confirmation.
The idea is of a piece of writing that connects the previous to the next one, as in a sequence of passages:
passage (n.) :
- early 13c., "a road, passage;" late 13c., "action of passing," from Old French passage "mountain pass, passage" (11c.), from passer "to go by" (see pass (v.)). Meaning "corridor in a building" first recorded 1610s. Meaning "a portion of writing" is from 1610s, of music, from 1670s.
- would therefore beg Leave to read a Passage where he explains himself, and plainly declares what his Thoughts are concerning the Indulgence granted by Law : This Passage is in the twentieth Page, where speaking of some wholesome Severities us'd in former Reigns, he adds, I would not here be misunderstood, as if I ...