In Byron's "She Walks in Beauty," hyperbole seems to achieve the purpose of allowing the reader to image how beyond the norm the author perceives the object of his poem to be...that she cannot even be compared to the grandeur of nature, in that even "starry skies" pale in comparison to this woman's beauty.
Hyperbole is defined as:
...obvious and deliberate exaggeration or an extravagant statement. It is a figure of speech not intended to be taken literally since it is exaggeration for the sake of emphasis.
In other words, hyperbole is used for effect: the author is exaggerating to draw attention the essence of his poem...that this woman is (we can infer) the most beautiful "creature" he has ever seen.
It is interesting to note that focus of the poem—Anne Wilmot—was Byron's cousin by marriage, and the poem was written when the poet saw her for the very first time.
According to literary historians, Byron’s cousin wore a black gown that was brightened with spangles.
It is noted, then, that the dark gown and "spangles" obviously brought to Byron's mind the dark night with stars glittering like diamonds. The seeming intent of hyperbole here, is to make the reader understand the sheer force of her beauty by comparing her to things that are incomparable:
She walks in beauty like the night
Of cloudless climes and starry skies...
Her beauty is also described as standing out beyond that of a sunny day—also an exaggeration.
Thus mellowed to the tender light
Which heaven to gaudy day denies.
The use of hyperbole as figurative language does not ask the reader to believe that the woman is really more beautiful than glittering stars on the back drop of the breath-taking skies of night, but to imagine that her beauty is extraordinarily powerful as the poet first lays eyes on her. The power is then conveyed by the poet's use of overstatement or exaggeration.