Said of Emperor Nero, last of the Caesars, who is partially responsible (but bears the bulk of the blame) for a 9 day fire that destroyed two-thirds of Rome in 64AD.
In this case the 'fiddling' is probaby figurative as fiddles weren't around in the first century. Instead it is probably an allusion to Nero's cavalier attitude to the fire, probably because it was his idea.
Nero wanted to rebuild much of the city of Rome that had been turning into urban sprawl over the years and decades. However, he ran into much resistance with the owners of the land and buildings that he wanted to tear down to start his urban renewal. Eventually the fire was halted by creating a firebreak by purposely burning still more buildings, and Nero did the political best he could: he blamed the Christians, who became societies scapegoat, and Nero still got to rebuild his city.
A more sinister (albeit probably wrong) explanation, building on the last, is that Nero set fire to Rome to see "how Troy would look when it was in flames and to serve as a suitable background for a recitation of his poetry while accompanying himself on the lyre" (Random House Dictionary of Popular Proverbs and Sayings).