Why is "hyperbole" pronounced so differently from its spelling?

Hyperbole comes from Greek ὑπερβολή, via Latin. When English adopts words from other languages, it often keeps both the spelling and pronunciation close to those of the origin language. Since other languages have different spelling conventions from ours — in particular, in many languages, a final e isn’t silent — many borrowed words have disparities like this: compare forte, mocha, jalapeno, etc.

(Another common cause of disparities between spelling and pronunciation is that spelling is much more resistant to change, so a spelling is often a fossil of an older pronunciation: that’s where things like the silent l in walk, talk come from. But that’s not what’s going on in this case.)

It's an example of a class of words that came into English from Greek, and retained a final "-e" as a separate syllable. Most such words are either names (Penelope, Calliope, Dione, Selene) or only in technical or learned use (synecdoche). Hyperbole is one that is slightly more widely used, which is why I think it has two competing pronunciations.

My guess would be that the final syllable was emphasized in the Greek as noted from its entry in NOAD.

ORIGIN late Middle English : via Latin from Greek huperbolē

So the bole would be more of a bolé than bowl

For what it's worth, it's composed of huper meaning over and ballō meaning to throw