It is a salute that has been associated with the Roman Empire for centuries, and enjoyed a resurgence in America in the late 19th Century, most notably appearing in the original Broadway version of Ben-Hur at around the same time it became used to salute the US flag. Ben-Hur exaggerated the gesture for the stage.

The popularity of Ben-Hur led to the exaggerated salute turning up in all sorts of silent films, including the 1914 Italian epic Cabiria, which earns its place in cinematic history for being the first film to ever feature a tracking shot, and in European history for having been written by the poet and right-wing iconoclast Gabriele d'Annunzio.

In 1919 d'Annunzio led a nationalist mob to occupy the disputed ex-Austro-Hungarian port of Fiume in the name of Italy. The Italian government, embarrassed by his action in light of their percieved inaction by the public - having been denied the port in the post-war divvying up of the old empire, and wishing to honour the negotiations for other ex-Austro-Hungarian territories that were in progress, declined to get involved.

Undeterred, d'Annunzio thus created the Italian Regency of Carnaro. To evoke the past glory of Rome and shame what he considered the disgrace of the Italian government (now blockading the port of Fiume to get him to surrender), he appropriated the Ben-Hur Roman Salute and devised other rituals such as huge speeches from balconies and chants. He dressed his thugs in black shirts. He used Castor Oil as a punishment He set up a form of government around corporatist lines, drawing policy from both right and left. He was also great at building populist support from residents of Fiume and of Italy. Very quickly he was ousted by Italian-backed forces and the Regency was over. d'Annunzio never reached such heights again, but the Regency provided the kernel around which the various Italian fascist movements grew.

Mussolini - who had never been able to get the press attention or notoriety of d'Annunzio, borrowed much of d'Annunzio's work while developing what became the National Fascist Party. The speeches, the populism, the chants and of course the Salute. Soon after the 1922 March on Rome which brought the Fascists to power, they instituted the use of the Salute in schools in the same manner it was being used in the US. The salute eventually replaced the handshake as the accepted greeting in Italian society.

At around this time, Nazi Party members took to using this Fascist salute. Hitler didn't discourage this, and it became a staple of the Nazi regime. Interestingly, there was a lot of disquiet from some quarters about a German nationalist movement using a foreign salute - from a country that had just humiliated Germany in the Great War no less. This led to Rudolf Hess writing an essay that played down Italian influence, that raising arms is the most natural thing, that the Nazi salute was authentically German. A large academic programme was then put into effect to scour German culture for a way to back that assertion up.

Sources:

Allert, The Hitler Salute: On the Meaning of a Gesture
Bosworth, Mussolini