Origins
The word 'sight' descends from Old English 'gesiht' or 'gesihþ' (vision, something seen, the faculty of seeing), from Proto-Germanic *ga-sihtiją, a noun derived from the verb *sehwan (to see) with the collective or perfective prefix *ga-. The verb itself traces back to PIE *sekʷ- (to see, to follow, to perceive), one of the most important sensory roots in Indo-European.
The Old English prefix 'ge-' (cognate with German 'ge-' as in 'Gesicht,' face/vision) was gradually lost in Middle English. By the thirteenth century, the form had simplified to 'sight' or 'sighte,' and the prefix was no longer felt as a separate morpheme. The German cognate 'Sicht' (view, visibility) preserves a similar form, while 'Gesicht' (face, vision) retains the prefix.
The PIE root *sekʷ- is remarkable for encoding both visual perception and physical following in a single concept. Through the Germanic branch, it produced 'see,' 'sight,' 'seer,' and the German 'sehen' (to see). Through the Italic branch, it produced Latin 'sequī' (to follow), which gave English 'sequence' (things that follow each other), 'consequence' (what follows from), 'prosecute' (to follow forward), 'persecute' (to follow through), 'subsequent' (following after), 'sequel' (what follows), 'second' (the one that follows the first), and 'suit' (from Old French, ultimately 'that which follows'). The conceptual bridge is intuitive: the eyes follow their object; a pursuer follows what they have seen.
Figurative Development
The compound words built on 'sight' are among the most expressive in English. 'Insight' (seeing into) means deep understanding. 'Oversight' has two contradictory meanings: supervision (looking over) and failure to notice (looking past) — a Janus word. 'Foresight' (seeing ahead) is prudence. 'Hindsight' (seeing behind) is retrospective wisdom, always perfect because the future has become the past. These compounds show the systematic metaphor that understanding is seeing — one of the most pervasive conceptual metaphors in English and many other languages.
The phrase 'at first sight' (upon first seeing) dates from the fourteenth century. 'Love at first sight' is attested from the sixteenth century in English, though the concept is far older — Greek 'erōs ek prōtēs opseōs' expressed the same idea. 'Out of sight, out of mind' — the proverb equating visibility with memory — is attested from the thirteenth century. 'Sightseeing' (visiting places of interest) dates from the eighteenth century.
The word 'site' (a location) is not related to 'sight' despite the similar pronunciation. 'Site' comes from Latin 'situs' (position, location), from 'sinere' (to leave, to place). The homophony is coincidental, though speakers have long confused and conflated the two, as in 'building sight' for 'building site.'