It doesn't. Italy was one of the most ravaged parts of the Western Roman Empire, it has less historical continuity than Spain or France. Only Britain does worse.
>If you go to Rome, the manhole covers read SPQR (Senatus Populusque Romanus).
That's propaganda. The link between the Roman state and Italy was something pushed by the fascists and it stayed behind in the popular mindset (since it's a more pleasant to think of yourself as the True Descendants of the Romans than a playground for the French, Germans, Arabs and Greeks). But go a few centuries back and people made as much a connection between Roman ruins and Italians as they do now between the Pyramids and modern Egyptians.
Realistically continuity goes back to the Renaissance, which saw the birth of an indigenous Italian culture as well as its mass export (favoured by the fact that albeit constantly invaded, Italy wasn't torn apart by the religious wars that destroyed the rest of the continent).