Why does a silent "-e" at the end of a word lengthen vowels?

The historical reason for this spelling pattern in general is most likely a process of vowel "lengthening" that applied in Middle English to certain words that previously ended in a schwa sound. This happened before the Great Vowel Shift, so that isn't directly relevant. Rather, the "Great Vowel Shift" is the explanation for why English "long a", "long i", and "long e" aren't similar to the sounds typically associated with the letters A, I and E in other European languages.

Word-final E was once used to indicate the sound [ə]

Word-final "silent e" wasn't always silent. In early stages of Middle English, word-final E was used to represent a schwa sound [ə] (which developed from various older vowel sounds). In late Middle English, this schwa was lost, but an originally short A, O or E in the preceding syllable ended up being pronounced long (as /aː/, /ɔː/ or /ɛː/, respectively, before the Great Vowel Shift.) An example of this change is the word knave [neɪv], from late Middle English [knaːv], from early Middle English [knavə], Old English [ˈknava].

Pronunciations with both a long vowel and final schwa (like [knaːvə]) probably existed at some point in Middle English, but I think the correct phonological analysis of such forms might be controversial. I've seen two distinct explanations proposed for the lengthening change: that it was conditioned by the vowel being in an "open syllable" (as well as some stress and syllable-position conditions), and that it was a form of "compensatory lengthening" where a vowel in a short syllable was made long because of the loss of the schwa phoneme in the following syllable.

It may also be relevant that "long" vowels existed in some loanwords from French that originally ended in schwa (spelled E in French, as in Middle English). French is certainly the source of the pronunciation of "long u" in modern English—the Old English "long u" sound came to be spelled as "ou".

After the loss of final [ə], word-final E developed other uses

Because of the sound change mentioned in the previous section, by the end of Middle English the spelling pattern -VCe ended up being used for many words that contained "long" vowels before a single word-final consonant. This caused final E, which no longer represented a vowel sound in and of itself, to become generalized as a marker of a "long vowel" before a preceding consonant letter.

There are many modern English words with this spelling pattern that have long vowels for other historical reasons, not because of the Middle English pre-schwa lengthening sound change; e.g. the words ice and wife had long vowels already in Old English (and didn't even end in schwa etymologically, although I don't know if they might have had schwa-final pronunciations at some point in Middle English during the time period when pronunciations with word-final schwa varied with pronunciations without the schwa sound).

And on the other hand, in most words with short vowels, spellings with word-final "silent e" eventually fell into disuse. So for example, the word frog, from Old English [ˈfrogga], probably ended in the sound [ə] before the loss of word-final schwa. A common spelling in Middle English was frogge, with final E, but this spelling of the word is now obsolete. In modern English spelling, "silent E" is only occasionally found after consonant clusters or double consonants; in a few words where it appears in this context, such as giraffe, it may be analyzed as a means of indicating that the final syllable is stressed.

Another use of "silent E" in Modern English spelling is as a marker of a preceding voiced fricative. The letter V rarely occurs word-finally in English; it is usually accompanied by word-final E, even in words like live (v.) and give that have "short" vowels. At a certain point in history, the sound /v/ did not occur word-finally in English; the existence of word-final /v/ in modern English can be attributed to the loss of word-final schwa, the introduction of loanwords, and some relatively recent clippings such as improv and maglev. Spellings with final V do occur for clippings and some loanwords.

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In conclusion, knowing that any particular word is spelled with "silent e" at the end doesn't tell you anything for sure about either that word's current pronunciation, or its historical pronunciation. The (imperfect but real) correlation between this spelling pattern and the use of a "long vowel" in the pronunciation of a word has a historical explanation in the Middle-English sound change that lengthened certain vowels in words that originally had word-final schwa.