A cursory search shows that the English adjective "wise" and the German verb "wissen" descend from the same root: the PIE *weyd- ("to see, to know"). I found this by using Etymonline to search the English word, and Wiktionary to search the German word. I trust the former, but not necessarily the latter. Could you kind folks confirm that this common ancestry is true? And if so, how did the phonetic changes come about?
These words are indeed cognate! They both stem from the PIE root *weyd-, meaning "to know" or "to perceive", along with less obvious cognates like "guide", "vision", and "eidolon".
In Proto-Indo-European, the vowels *e, *o, and *[nothing] tended to alternate, a pattern called "ablaut". You can see traces of this still in the "strong verbs" which change their vowels to indicate tense, like drive/drove or freeze/froze. So the "zero-grade" form of *weyd- was *wyd-, also written *wid-.
In Proto-Germanic, *wyd- gave rise to a verb *witaną; the *d became *t through Grimm's Law.
In Old High German, the *t then lenited into a z, giving OHG wizzen; z then developed into s, giving Modern German wissen. (Somewhere along the line <w> became pronounced as /v/, though the spelling remained.)
In English, the *t remained, but the inflectional endings slowly dropped off, giving Old English witan, and Modern English wit. The verb died out but the noun survived. (An inflected form of this, with a mutated vowel, was generalized to give the poetic verb wot "know".)
One of the participles of *weyd- was *weyd-tos, which became *weyd-stos and then *wī-saz in Proto-Germanic. Again in English the endings dropped off, giving Old English wīs, which was respelled wise in Modern English. The pronunciation of the vowel changed in the Great Vowel Shift and the /s/ lenited to give the modern /wajz/.
Yes, en wise and de wissen share a root, but not as recently as Levenstein distance would suggest.
de weise is more directly cognate with en wise, whereas de wissen is directly cognate with en wit, which until recently was also a verb.
Yes, not coincidentally, wise and wit also share a root, but it split before the ancestor of English and German split.
The confusion is that t changed to something like s or tz in many words in the ancestor of High German and related languages, but not in other Germanic language families.
The 'i' being pronounced differently in English is because of the 'great vowel shift' (look it up). As for the w being pronounced differently, its because the w was originally pronounced as the English w in all Germanic languages, but in several of them (including German), it changed into a v sound.
Also, the final 'e' in English 'wise' was originally pronounced. This caused the 's' to be pronounced as a 'z' sound, which is a common thing to happen to consonants that appear in between vowels. This obviously didn't happen in German though. Funny enough, the 's' in German is more often than not pronounced as a 'z'. Its only pronounced as an 's' when it appears at the end of a syllable or word, or its doubled (as in wissen).
As for what the word they are descended from meant, I don't know. But its obvious that these two are related. Wissen means 'to know (a fact)' while 'wise' means something like 'to be knowledgeable'. Not only are their meanings related, but they also obviously resemble each other. German is one of English's closest surviving relatives (though not the closest, English's closest relatives would actually be the Frisian languages).