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To observe. Twig the cull, he is peery; observe the fellow, he is watching us. Also to disengage, snap asunder, or break off. To twig the darbies; to knock off the irons.
Definition taken from The 1811 Dictionary of the Vulgar Tongue, originally by Francis Grose.
In Twig * TwissNathan Bailey's 1736 Dictionary of canting and thieving slang
John S. Farmer's collection of canting songs and slang rhymes
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Francis Grose was independently wealthy, having inherited money from his father, a jeweller. Finding himself overspending, he published a number of books; his Provincial Glossary seems to have been the starting-point for the Vulgar Tongue reproduced here.