I love this question! And like most good questions, it has several interrelated answers! It's almost hard to know where to begin!
First, it might be good to remember that the regime in France at the time was known as the Second Empire, and the country was ruled by a man named Napoleon III, nephew of Napoleon I. Because the Bonaparte family was not a legitimate ruling house (they came to power during the French Revolution of 1789 and made their name as military conquerors / statesmen), they were constantly looking for ways to bolster their legitimacy in the eyes of the French people and other European powers. One way of doing so was marriage alliance. Napoleon III's cousin, named Napoleon-Jerome Bonaparte, was thus married to a daughter of Victor-Emmanuel II, King of Sardinia-Piedmont, later first King of Italy, and heir to an ancient dynasty.
In addition to securing this dynastic alliance, Napoleon III was interested in living up to family expectations set by his uncle, the famous general Napoleon I, who conquered most of Europe between 1796 and 1814. The former was not nearly as talented as the latter, though, so he relied on diplomacy to conquer new territories for France. An important part of his support for the unification of Italy under Victor-Emmanuel II was that two culturally French territories ruled by the Kingdom of Sardinia-Piedmont -- namely, Savoy and Nice -- be ceded to the Second Empire.
Finally, it might be noted that France was a traditional enemy of the Habsburg Empire, which was the dominant military / political power on the Italian peninsula through the 1850's. Supporting the aspirations of Victor-Emmanuel II, who wanted to unify the peninsula under his rule, which meant of course driving out the Habsburgs, was a terrific way to advance France's diplomatic goals and international standing.
I hope these answers help!