Not to discourage other answers but this questions was answered previously by u/listyraedsar here.

The salute dates back to a painting by Jacques-Louis David's called the Oath of the Horatii. It depicts three young men with their arms outstretched pledging allegiance to Rome. This idea of a "roman salute" spread quickly and made its way into many different mediums, until the salute caught the eye of the Italian fascist party, and from that it spread to the nazis.

Interestingly enough there is no real proof that Romans actually saluted like that, David essentially made it up.