Perfectly fit; in excellent health; in excellent condition.
My aunt is 58, has had 6 children, but she is as fit as a fiddle.
The word ‘fiddle’ here is the colloquial name for violin. ‘Fit’ didn’t originally mean healthy and energetic, in the sense it is often used nowadays. When this phrase was coined ‘fit’ was used to mean ‘suitable, seemly’, in the way we now might say ‘fit for purpose’.
Thomas Dekker, in The batchelars banquet, 1603 referred to ‘as fine as a fiddle’:
“Then comes downe mistresse Nurse as fine as a farthing fiddle, in her petticoate and kertle.”
Not long afterwards, in 1616, there’s W. Haughton’s English-men for my Money, which includes:
“This is excellent ynfayth [in faith], as fit as a fiddle.”