set off v.

Middle English setten, from Old English settan (transitive) "cause to sit; make or cause to rest as on a seat; cause to be put, placed, or seated;" also "put in a definite place," also "arrange, fix adjust; fix or appoint (a time) for some affair or transaction," and "cause (thoughts, affections) to dwell on."

This is from Proto-Germanic *(bi)satejanan "to cause to sit, set" (source also of Old Norse setja, Swedish sätta, Old Saxon settian, Old Frisian setta, Dutch zetten, German setzen, Gothic satjan), causative form of PIE *sod-, a variant of the root *sed- (1) "to sit." Also see set (n.2). It has been confused with sit (v.) at least since early 14c. 

The intransitive sense of "be seated" is from c. 1200; that of "sink down, descend, decline toward and pass below the horizon" (of the sun, moon, or stars) is by mid-13c., perhaps from similar use of the cognates in Scandinavian languages; figurative use of this is from c. 1600.

Many uses are highly idiomatic, the verb, like put, its nearest equivalent, and do, make, get, etc., having become of almost universal application, and taking its distinctive color from the context. [Century Dictionary]

The sense of "make or cause to do, act, or be; start, bring (something) to a certain state" (on fire, in order, etc.) and that of "mount a gemstone" are attested by mid-13c. That of "determine upon, resolve" is from c. 1300; hence be set against "resisting" (mid-14c.).

The sense of "make a table ready for a meal" is from late 14c.; that of "regulate or adjust by a standard" (of a clock, etc.) is from late 14c. In printing, "to place (types) in the proper order for reading; put into type," 1520s. From c. 1500 as "put words to music." From 1570s as "put (a broken or dislocated bone) in position." In cookery, plastering, etc., "become firm or solid in consistency" by 1736.

To set (one's) heart on (something) is from c. 1300 as "love, be devoted to;" c. 1400 as "have a desire for." To set (one's) mind is from mid-15c.; transitive set (one's mind) to "determine to accomplish" is from late 15c.  To set (something) on "incite to attack" (c. 1300) originally was in reference to hounds and game. To set an example is mid-14c. (set (v.) in the sense of "present" is from late Old English). The notion of "fix the value of" is behind old phrases such as set at naught "regard as nothing."

To set out is from c. 1300 as "display (for sale);" to set up shop "commence doing business" is from c. 1400.

by c. 1200 as an emphatic form of Old English of (see of), employed in the adverbial use of that word. The prepositional meaning "away from" and the adjectival sense of "farther" were not firmly fixed in this variant until 17c., but once they were they left the original of with the transferred and weakened senses of the word. Meaning "not working" is from 1861.

Off the cuff "extemporaneously, without preparation" (1938) is from the notion of speaking from notes written in haste on one's shirt cuffs. In reference to clothing, off the rack (adj.) "not tailored, not made to individual requirements, ready-made" is by 1963, on the notion of buying it from the rack of a clothing store; off the record "not to be publicly disclosed" is from 1933; off the wall "crazy" is 1968, probably from the notion of a lunatic "bouncing off the walls" or else in reference to carom shots in squash, handball, etc.