Millar doesn't expound the semantic shift at all, but Wiktionary tries to. But wið is a functional morpheme, that at large change with less probability. So why did it shift "to denote association rather than opposition, from senses of attaining proximity through movement towards and then rest at a thing, displacing Middle English mid (“with”), from Old English mid (“with”)"?
A few words are easy, especially the little grammatical ones: her is ‘here’, and is ‘and’, æfter is ‘after’, þær is ‘there’, his is ‘his’, þæt is ‘that’, and him is ‘him’ – at least sometimes! Only slightly harder are cyning and its contracted form cyng ‘king’, rad ‘rode’, niht ‘nights’ and wolde, woldon ‘would’. And you have probably spotted that sæt is just our word ‘sat’. Barely recognizable is aðas ‘oaths’, but, if you ignore the prefix ge-, you can see that gefeaht is the same word as our ‘fought’. You may be startled to learn that the mysterious-looking ealne is just our word ‘all’ with a grammatical ending attached. Finally, that word wið is just our word ‘with’, but note that the word meant ‘against’ in Old English. The Old English word for ‘with’ was mid, which has completely disappeared except in the compound ‘midwife’ (literally, ‘with-woman’); its job has been taken over by wið, which in turn has handed over its original meaning to yet another word, against, except in the ambiguous phrase he fought with his brother and in the verb withstand, which has a similar meaning to ‘stand against’.
Revised by Robert McColl Millar, Trask's Historical Linguistics (2015 3e), p 3. Trask died in 2004.
The items in the list are all (supposedly) of the sort I called basic vocabulary in Chapter 2. That is, they are words that change more slowly than vocabulary in general: pronouns, low numerals, body-part names, simple verbs and adjectives, and so on, although it is important to note that, as Dixon (1997), among others, points out, and we have already discussed in Chapter 2, some cultures may alter ‘central’ items of vocabulary regularly, in order to avoid taboo words and names.
Op. cit. p 352.
Etymology 1
From Middle English with, from Old English wiþ (“against, opposite, toward, with”), from Proto-Germanic *wiþi, a shortened form of Proto-Germanic *wiþrą (“against”), which see for its derivation and other descendants cognate to the English preposition. In Middle English, the word shifted to denote association rather than opposition, from senses of attaining proximity through movement towards and then rest at a thing, displacing Middle English mid (“with”), from Old English mid (“with”), from Proto-Germanic *midi; an earlier model of this meaning shift exists in cognate Old Norse við; elsewhere, the converse meaning shift is exposed by Old South Arabian _𐩨𐩺𐩬_ (byn, “between, amid”) spawning Old South Arabian _𐩨𐩬_ (bn, “against”) and even likewise frequent reverse meaning _𐩨𐩬_ (bn, “from”).