Middle English sēn, from Old English seon (Anglian sean) "be or become aware of by means of the eye; look, behold;" also "perceive mentally, understand; experience; visit (a place); inspect" (contracted V strong verb; past tense seah, past participle sewen), from Proto-Germanic *sehwanan (source also of Old Saxon, Old High German sehan, Middle High German, German sehen, Old Frisian sia, Middle Dutch sien, Old Norse sja, Gothic saihwan).
This is reconstructed to be from PIE root *sekw- (2) "to see." That PIE root often was said to be probably identical with *sekw- (1) "to follow," which produced words for "say" in Greek and Latin, and also words for "follow" (such as Latin sequor), but "opinions differ in regard to the semantic starting-point and sequences" [Buck]. Thus see might mean, etymologically, "follow with the eyes" (and in some languages extending to "speak, say, tell"). But OED finds this "involves a hypothetical sense-development which it is not easy to accept with confidence," and Boutkan also doubts the connection and gives the word "No certain PIE etymology."
It is attested by late Old English as "be able to see with the eyes, have the faculty of sight, not be blind."
As the sense of sight affords far more complete and definite information respecting external objects than any other of the senses, mental perceptions are in many (perh. in all) languages referred to in visual terms, and often with little or no consciousness of metaphor. [OED]
English see has been used in many of these senses since early Middle English: "foresee; behold in the imagination or in a dream," also "to recognize the force of (a demonstration)," all c. 1200.
It is attested by c. 1300 as "ensure, make sure" (something is so, someone does something). To see to is by late 14c. as "be attentive to, take special care about" (also "to look at"); hence "attend to, arrange for, bring about as a result." See to it "take special care; see that it be done" is from late 15c.
The sense of "escort" (as in see you home) is attested c. 1600 in Shakespeare. The meaning "to receive as a visitor" is attested from c. 1500. The wagering sense of "equal a bet, accept by staking a similar sum" is by 1590s. Used in phrases expressing comparative and superlative (best I've ever seen) from early 14c.
Imperative use of see! "look! behold!" is by early 14c. Emphatic expression see here is attested from early 15c.; probably the notion is "see, here is ...;" but the modern use of it as "a brusque form of address used to preface an order," etc. [OED] is by 1897 in schoolboy talk. The qualifying expression as far as I can see is attested from 1560s.
Let me see as a statement expressing consideration when the speaker is trying to recall something is recorded from 1510s. See you as a casual farewell is attested by 1891 (see you soon; probably short for hope to see you soon). To see something in (someone, etc.) "perceive good or attractive qualities in" is by 1832.