1) The referendum in Veneto was supported by the Venetian section of all major parties, across all the political spectrum.
2) In Italy five regions have a far greater degree of autonomy for historical reasons. Many Comuni (city councils) voted to leave Veneto to join nearby Trentino Alto-Adige just for this far greater autonomy. The proponents of the referendum would like to see more federalism in Italy, e.g. following the German model.
3) Italy is very divided from an economical point of view. Veneto and Lombardy may be easily compared to the richest German industrial districts, whereas the South's economy is not so different from Greece. Taxation in Italy is very high, and a lot of money gets redistributed from the rich North to the poorer South. To make a long story short each citizen from Veneto gives about 5000€ per year to each citizen in the South. Edit: see noisymemories's comment below, often the money is lost in a maze of entities and inefficiency.
I perfectly acknowledge a country should redistribute wealth, but many perceive that this redistribution is too high and unfair, and call for a less centralized Italy.
4) Italy is a young country, with 150 years of history, whereas Veneto has millenium-long story as the Serenissima Republic. The Venetian language is widespread alongside Italian, and many (again, across all political spectrum) have stronger ties to the Venetian heritage rather than to the Italian heritage.