How did 'to wit' shift (from "to know") 🡺 to mean 'that is to say'?
Unquestionably, "wit" or "knowing" are concepts distinguishable from "saying". Thus how did 'that is to wit' shift 🢂 to denote 'that is to say; namely'?
Both the noun wit [OE] and the verb [OE] go back ultimately to the Indo-European base *woid-, *weid-, *wid-. This originally meant ‘see’, in which sense it has given English visible, vision, etc, but it developed metaphorically to ‘know’, and it is this sense that lies behind English wit. The noun to begin with denoted ‘mind, understanding, judgement, sense’ (a meaning preserved in expressions such as ‘keep one’s wits about one’ and ‘slow-witted’), and the modern sense ‘clever humorousness’ did not begin to emerge until the 16th century. The verb has now virtually died out, except in the expression to wit. Witness is etymologically the state of ‘knowing’. Other English words that come from the same Indo-European base or its Germanic descendant include guide, history, idea, story, and twit.
John Ayto, Word Origins (2005 2e), p 549 Left column. Below I quote Etymonline on the verb "wit":
"to know" (archaic), Old English witan (past tense wast, past participle witen) "to know, beware of or conscious of, understand, observe, ascertain, learn,"
from Proto-Germanic *witanan "to have seen," hence "to know" (source also of Old Saxon witan, Old Norse vita, Old Frisian wita, Middle Dutch, Dutch weten, Old High German wizzan, German wissen, Gothic witan "to know"), from PIE root *weid- "to see."
The phrase to wit, almost the only surviving use of the verb, is first recorded 1570s, from earlier that is to wit (mid-14c.), probably a loan-translation of Anglo-French cestasavoir, used to render Latin videlicet (see viz.).