An exclamation made to indicate "nonsense; rubbish".
The term fiddlesticks derives from the literal 'fiddle sticks', that is, the bows that are used to play violins. Those have been named in English since the 14th century - then as 'fydylstyks'.
The word was appropriated to indicate absurdity in the 17th century. Thomas Nashe used it that way in the play Summer's Last Will and Testament, 1600:
A fiddlesticke! ne're tell me I am full of words.
There's nothing inherently comic about a violin bow. It seems that 'fiddlestick' was chosen just because it sounds like a comedy word, like 'scuttlebutt' (a cask of drinking water), 'lickspittle' (a sycophant) and 'snollygoster' (an unprincipled person).
In the same way the 'I don't give a fig' was originally 'I don't give a fig's end', that is, it referred to something insignificant, 'fiddlesticks' was originally 'fiddlestick's end', that is, it was a reference to something paltry, trifling and absurd.