How did the comparative of "well" become "better"?

An interesting question. First, this is called suppletion (or suppletive forms) when a word has different forms that come from different roots. You see it in the paradigm for be (am, are, is, was, were, being, been) and in go (goes, went, going, going). Went was originally the past form of wend. I have never read anything on why suppletion happens, and like most of language change I think it would be nearly impossible to say, but we can often trace these changes.

Now to good and well. In Present-Day English, most adverbs are formed from adjectives by adding the suffix -ly. This morphological process stems from the Middle English periuod. In Old English adjective were regularly formed by adjectives by the addition of the suffix -e (fæger 'pretty, beautiful' (whence [i[]fair[/i]), fægre 'pleasantly, beautifully'). The suffix -lic was in Old English adjectival and there was a corresponding form in -e to make adverbs from adjectives, -lice. Final e soon became unstressed schwa and finally silent, so the two suffixes were sounded the same, and then got reanalyzed as purely an adverbial suffix. OE wel comes from the same root in Proto-Germanic as the auxiliary verb will.

Now part of the story is why we use adverbs to qualify verbal action, when most Germanic languages use plain old adjectives. In German is is grammatical to say schlaff gut 'sleep well'. Well is also an adjective in English: I am well. I hope that helps.

Ceci n'est pas un seing.