Two immediate lexical progenitors produced transitive use of the colloquial phrasal verb 'to pull off' in the sense of
2. trans. colloq. To succeed in accomplishing, achieving, or producing (something); to carry off.
OED, 'to pull off'
Those two progenitors were uses in these senses:
1. trans.
a. To take away or detach (something) by pulling from where it is held or attached. ....
b. To take off (one's coat, etc.); to doff (one's hat).
op. cit.
Senses 1a and 1b are attested in OED from before 1425 (citation composed before 1399) and around 1500, respectively. Sense 2 is attested from 1860:
Baily's Monthly Mag. 1 34 After the good old matches of Club and Ground against Cambridge and against Oxford are pulled off at the two Universities, the London Season will open.
Although pulling off boots and coats, etc. (1b), often has a sudden, tangential element of success, it is likely that, at least in the US, transitional, figurative political uses played directly into the ready adoption of the sense denoting successful accomplishment, achievement or production.
A painstaking examination of primary sources — uses in the popular press — showed continuing dominance of sense 1b into the mid-1800s (lots of clothes were pulled off during that period), along with, much less frequent, uses in sense 1a (often with reference to pulling off parts of plants in articles concerning agriculture and gardening; sometimes with reference to pulling off flesh and skin).
The principal difference between senses 1 and 2 is relative materiality. Sense 1 takes tactile material as its object: boats, clothes, vegetation, flesh. The later sense 2 takes ideas or concepts as its object: a win, a success, an accomplishment, an achievement. Such chronological development of lexical sense is a standard process; it goes alongside the usually less-definable adoption of figurative senses.
Some transitional US uses, notably political, illustrate the development of the sense, from material things pulled off to immaterial, by way of figurative uses:
The Democratic Journals in New York have a mournful look. Fillmore, on one side, pulls on the anti Ostend conservative men, and Fremont, on the other, pulls off the Germans, Irish, &c., ....
The Wilmington Daily Herald (Wilmington, North Carolina), 18 Jul 1856.
Three times he has left governmental posts, ...after brave displays of political pluck; and now, for the fourth time he pulls off his cabinet cloak and throws it in the faces of his old fogy associates.
True American (Steubenville, Ohio) 08 Mar 1855.
These early, figurative US uses from 1855 and 1856 precede the earliest US use of 'pull off' in sense 2 found, from 1858:
His crowning success, however, was to pull off the two thousand guineas stake with Fitz Roland and the Derby with Beardsman.
The Brooklyn Daily Eagle (Brooklyn, New York) 05 Jun 1858.
As will be deduced from the currency of the stakes, if nothing else, that 1858 use, although it appears in a US newspaper, refers to an English horserace.
Setting aside the US transitional, figurative uses of 'pull off' in political contexts, and turning to UK uses, evidence of the development of transitive sense 2 appears earlier. In 1851, for example, rhetorically linked uses of the intransitive verbal phrase 'came off' and the transitive 'carry off' prepare for a third with the transitive 'pull [something] off'. The general subject is horse racing, and the direct object is an event:
The opening race across the flat came off in favour of the four-year-old Triennial winner...we suggested the probability of some undreamt of outsider carrying off the prize...[w]hether he has won or not, old Joe Rogers will delight in the accomplishment of the feat, for he...abhors pulling an event off with a favourite.
The Era (London) 19 October 1851
A slightly (four days) earlier use in scare quotes suggests the application of 'pull off' to the immaterial object of a horse-racing title is a neologism...at least as concerns the author of the piece and his judgement of Dublin readers' familiarity with the use of the phrase:
...and Mr. Quin's Whiff "pulling" off the First Class of Trainers' Stakes.
Freeman's Journal (Dublin) Wednesday 15 October 1851
Here again, in the UK, uses of 'pull off' in sense 2 are presaged by figurative uses in political contexts in sense 1b; such figurative use again involves the removal of articles of attire. One such, a very early instance, is this citation from the OED sense 1b attestations:
1677 W. Hubbard Narr. Troubles with Indians New-Eng. (new ed.) ii. 32 He pulled off his Vizour of a friend, and discovered what he was.
Another, contemporaneous with (although a few days earlier than) the 1851 uses in horse-racing contexts:
If the Roman Catholic Prelates...will come forward to denounce...the doings of Italian tyrants...we shal sing our palinodia with good will...[t]ill than, we shall continue to regard them as accomplices of the Emperor of Austria and Marshal Haynau...as so many MacHales and Cullens in disguise, waiting for the suitable moment to pull off the mask, draw the sword, and throw away the scabbard. — The Patriot
Belfast Mercury 02 October 1851
Figurative uses with reference to the pulling off of masks were thematic in the evidence I examined pertaining to sense 1b.
Side Note on 'Stake' and 'Stakes'
It may be objected to the foregoing account that a 'stake' is material: valuable goods or money wagered on the outcome of a bet, game, or event.
So, then, uses of 'pull off' with, as direct object, the 'stake' wagered on a horse race, must at least be said to be figurative uses in sense 1a ("to take away" something "by pulling from where it is held"), rather than sense 2 ("to succeed at something" or "to win something").
However, the citations shown make clear that, where 'stake' or 'stakes' is used as the object of 'pulling off', it is being used in a specialized sense, and denotes the race (singular) or class of race (plural), rather than the material prize. The sense is OED sense 3a of stake, n.2:
pl. in Horse Racing, Coursing, etc., the sums of money staked or subscribed by the owners who enter horses or dogs for a contest, the whole to be received as the prize by the owner of the winner or divided among the owners of the animals ‘placed’, as declared in the conditions of the contest. Hence in sing. (cf. SWEEPSTAKE n.) a race for money thus staked or subscribed. Also in pl. with defining words as the designation of particular races or classes of races in which the sum of money staked is the prize as distinguished from a Plate (see PLATE n. 4), Cup, or the like.
(Bold emphasis mine.)
Thus, where the direct object of 'pull off' shown in the citations is not the "match" (1860, Bailey's Monthly Mag) or the "event" (1851, The Era), but rather "the two thousand guineas stake" (1858, The Brooklyn Daily Eagle) or the "First Class of Trainers' Stakes" (1851, Freemans Journal), the denotation of 'stake' is the race and the denotation of 'Stakes' is the class of race.
Best Answer
Dictionaries
Partridge (2002) gives a definition and suggests a year but no explanation:
The OED collects a number of slang and colloquial phrases that use sock. In one's socks is a measure of a person's stature; to knock the socks off is to trounce another; to pull up one's socks is to make an effort.
No etymology is given for the phrase but their first three quotations are:
More examples
It also appears in Sir Arthur Conan Doyle's The Stark Munro Letters (1895):
Here's an example from Oliver Osborne's In the Land of the Boers (first published 1893, this revision 1900) (read online):
A when and where but not why can be found in the January 20th, 1900 edition of Notes and Queries (read online) amongst other sock phrases:
The March 7th, 1906 Punch magazine (read online) contains the same text as the OED's Daily Mail 1906 citation. Here's a fuller quotation from Punch:
And, after reporting another Smart Set phrase had been heard in the East End: