Mazzini believed Italy should be united as one country because all Italians shared a common nationality and cultural heritage.
Details:
Giuseppe Mazzini (1805-1872) was a powerful promoter of Italian nationalism during the 19th century, at a time when the Italian peninsula was divided into many smaller entities. He founded the group known as Young Italy and was a major figure in the Italian unification movement known as "Risorgimento" ("rising again"). He was a strong voice calling for all Italians to unite together in a republic. Let me quote you a bit from Mazzini, an example of his persuasive style:
Italy did not finish its nationalist program of unification as soon as Mazzini would have wanted. A united Italy finally came together in 1871, not long before Mazzini's death. But he had been a pioneer of the movement that brought Italians together during his lifetime.