Meaning:
Something that makes no sense.

Origin:
This line originates in Shakespeare's Comedy of Errors, 1590:

DROMIO OF SYRACUSE:

Was there ever any man thus beaten out of season,
-> When in the why and the wherefore is neither rhyme nor reason?




The bard must have liked the line as he used it again in As You Like It, 1600:

ROSALIND:

But are you so much in love as your rhymes speak?

ORLANDO:

Neither rhyme nor reason can express how much.