There’s no major connection between the Wild West and Italians that I can think of, and the major Italian film genre that preceded it, neorealism, was quite somber and serious. Seems random.
There had been the odd Italian western since the silent days and several just preceding the famous Spaghetti westerns, so they didn't come as completely out of the blue as it might appear, they just hadn't been as internationally visible. The Western genre had always been popular throughout Europe, in many media, and some of the Italian examples were co-productions with Spain or Germany.
Incidentally, the neo-realist genre may have been the one most apparent to international critics but Italian cinema covered a wide variety of genres and the most relevant other popular genre immediately preceding the well-known Westerns were the Sword and Sandal films (one of which was co-directed by Sergio Leone), These, some starring Steve Reeves, to some extent introduced Italian popular cinema to the world and paved the way for acceptance of the Italian Westerns.
As soon as A Fistful of Dollars broke out into international popularity and profitability, helped in no small measure by Clint Eastwood's TV popularity as well as Leone's fresh take on the genre, then the floodgates opened and many more followed.