While a good answer has been linked below, you might be interested in the actual mechanics of the Norman conquest of Sicily and Southern Italy. I wrote an answer here which you might be interested in. You might also be interested in the first part of this other longer answer narrating sociopolitical development in pre-unification Southern Italy also extensively examines the Norman conquest.

The short answer is that ethnic Normans were already a leading component of the armed forces of the Lombard Principality of Salerno, and had a pivotal role the stabilization of that principality in the late tenth century. Norman leaders, brothers William, Roger, Drogo, and Robert, all from the town of Hauteville in Normandy, carved for themselves a semi-autonomous fief in the Italian region of Puglia, which they had seized from the Byzantine Empire in the name of Prince Guimario of Salerno (Guimario, like all Lombard Princes, was in perpetual conflict with the Empire). So rather than an outright conquest, it's more correct to say that ethnic Normans integrated themselves in the martial/land-based political hierarchy in Southern Italy. Without diminishing the astounding military achievements of individuals, they nonetheless first built a social and economic power base, and only in a second moment neutralized the disunited and dysfunctional local political class across three political organizations: the Byzantine Theme, the old Lombard princes (once their initially positive relationship broke down), and the Sicilian Emirs.

But why Normans? Of all the foreign mercenaries to pass through the port of Salerno (for the existence of that harbor and entrepôt is the great environmental determinant of the Lombard Princes) why were the Normans the ones who stayed? And the answer is that they weren't the only ones: military recruitment in the Principality also targeted Northern Italian Lombards, Britons, and Provençals in addition to the Normans. But it would be the Normans who would assume positions of leadership, this perhaps by pure chance (maybe fraternal determinants were a greater advantage) or perhaps because conditions in Normandy made chain migration desirable. In all though, the important thing is that it would be this Norman leadership which would unify Southern Italy, and that same leadership would go on to conquer the Island of Sicily, and finally construct a polity that would be (for a few hundred years at least) one of the most formidable in the Mediterranean in both terms of pure projection of state power as well as social and economic complexity.

Hello, I think you’ll be interest in this post by u/AlanwithTea. It goes into detail about how a branch of Normans ended up in Sicily/Italy, and what their connection was to the northern branch was like.