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Undeniably, "savoir" and "dire" are different concepts. Thus how do they appertain to each other? The same semantic shift also happened in English, from "that is to wit" to "that is to say".

wit

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.).

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    "C'est à savoir" ne signifie pas "C'est à dire". C'est "A savoir" qui signifie "C'est à dire".
    – XouDo
    Commented May 27, 2021 at 7:47

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I do not think one can dissect phrases and try to reconstruct the meaning of each word that builds them. Phrases should be considered as new entities altogether, and they often are since dictionaries often have distinct entries for them. So I do not agree in saying savoir and dire "appertain to each other" semantically.

Savoir has a lot of different meanings. In the phrase à savoir the meaning implied is that of connaitre, but even then, as is often the case with phrases and, as stated in Le Robert the original meaning of the verb is weakened.

When I use à savoir I intend to explain something, to make something known to the person I am talking to.

In c'est-à-dire the meaning of dire is expliquer, as found in the Dictionnaire de Furetière1. When I use c'est à dire I intend to give an explanation to the person I am talking to.

I would venture to say that c'est-à-dire could be seen from the angle of the one who is explaining, and à savoir from the angle of the one who is receiving the information.

1 DIRE, signifie encore simplement, Expliquer. Cet endroit est obscur, mais l'Auteur veut dire que.

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