The short answer is Propaganda, Political intrigue and foreign/incompetent domination. And luck. Tons of luck.

Italians enjoyed a really short period of unity during the napoleonic wars: but this short, revolutionary moment of unity and freedom from old and incompetent monarchies, sparked a new wave of ideas, consolidated the ideals that were started in the coffehouses by italian enlightment philosophers, and spread them all over the place, unifing many concepts and cultural beliefs of the "different" italian cultures.

Most of all, this period saw the unification of the intellectual and classic-oriented (greek and roman culture) middle class, that saw the benefits of a united Italy and a comeback to a national entity as inevitable and glorious.

Then the Congress of Vienna happened and all the good that the unification showed were just trashed in a couple of months, with the uneventful return of the old monarchies and the foreign influence of the Austrian Empire in Northern Italy.

This is when republican, indipendentist and unification movements began to ran throughout the states in the Italian Peninsula, claiming a unified country and a competent government agaisnt the ancient regimes.

Obviously, these movements were mainly moved by the intellectual middle class, but soon had the capacity to bring under their wings several masses from the lower classes, both talking about freedoms, better government, and using blatant propaganda: In this period there was a flourishing of Operas, writings, novels, that talked about rebellion against the oppressor, about classical heroes and Italian History Characters that had to be followed as a good example for the perfect society;

Then of course there was Luck: in 1848 the entire European continent was thrown off by revolutionary movements, and Italy was no exception: Milan rised against the Austrians (5 days of Milan), showing to the word that Italians didn't want foreigners on their ground anymore and that they wanted to unify.

The one who saw through this and tried to obtain something out of it was the Kingdom of Sardinia-Piedmont, ruled by the Savoy Royal House, that tried invane to invade the Austrian dominated Lombardy in 1848 and had to fall back. But this was enough to show a good hope for a new beginning. Soon they reorganised, allied themselves with the French Empire and had their way for a second war: this time, with the backing of the French, the sardinians had their way through lombardy.

With the austrians defeated, several other small states decided to join Sardinia Piedmont, through pacific means; at this point just the Vatican and the Kingom of the Two sicilies stand.

Then Sicily rised against their king (there were several other revolts against the Bourbon domination, always failed) and a Sardinian general, Giuseppe Garibaldi, organized a "voluntary army" to offer help to the insurgents. In short, the "voluntary army" soon had indirect support from Sardinia and Direct support from the British, and started to take over the spared Neapolitan troops, conquering Sicily, and then climbing up the southern part of Italy, dethroning the bourbons creating a semi indipendent entity. At that point, the King of Sardinia descended to stop this situation and Garibaldi promptedly gifted him all his conquests, unifing the country in 1861 (plus Veneto in 1866 and Rome in 1870).

At this point the "big differences" were merely related to the lower classes. It wasn't really a "Culture clash", it was more of an economical clash. The north was slightly more industrialised and kept attracting internal and foreign investments, while the south didn't had so much luck ( and statal intervention) to get out of its economical and technical stagnation, keeping its economy to more rural and agrichultural basis.
The Middle and high classes, on the other hand, soon joined together, developing the basis of the common italian culture, and quickly spreading them around with the public institutions and schools.

The last chapter of unification for the Italian common people was the first world war: in the trenches of the Carso, millions of Italians joined together from all over the country, mixing their dialects, accents and cultures together in the struggle to survive one day more, finally sharing a common experience and a common ground upon which to built and consolidate all the ideas previously just kept in the fancy coffeehouses of the upper-middle class.

I know i digressed a bit, still Hope I answered well enough, if you have any doubt don't hesitate to ask.