There are two main factors that allowed Benito Mussolini and his fascists to come to power in Italy in the early 1920s.
The first of these factors stemmed from World War I. In that war, Italy had participated on the side of the victorious Allies. Italy had joined the war somewhat late after making a secret treaty with the Triple Entente. This agreement called for Italy to get a great deal of territory from Austria-Hungary after the war. When the war ended, however, the other allied powers did not end up giving that territory to Italy. This left Italians very angry with the status quo after the war. This discontent made it more likely that a virulently nationalist party like the fascists would be successful.
The second factor was economic and social. In the years after WWI, the Italian economy was beset by high unemployment and other economic problems. This led to social unrest. There were massive strikes that contributed to a sense of chaos and uncertainty. During this time, communists in Italy became much more numerous and some worried that they might take the country over.
In this climate, it was possible for fascism to rise. Fascists promised a stronger Italy that would be respected. They also promised to do away with the communists and to ensure labor peace and general order. This was just what Italians of this time period wanted to hear and so they supported Mussolini and his party.