The short answer is that Italian communist networks played a huge role in partisan resistance to fascist rule, both directly against Mussolini's government and then against the Nazi puppet regime of the Republic of Salo after the south of the country was liberated. Italy's geography (lots of mountainous regions where bands of partisans could hide) and the relatively early successes of the Allied invasions (compared to France for example) allowed for the partisans to form a bigger bastion of support for the communist party after the war. It also helped that partisan groups had been operating significantly longer than in France and were less dominated by a government in exile that wasn't far left (de Gaulle).

Additionally, Italy was less centralised than other European countries with a long history of cities/regions with very independent local political cultures, and was very recently unified as a nation state compared to other parts of Europe (the Risorgimento was less than a hundred years before WW2) - so you have certain areas that were absolute strongholds of communist support such as Bologna and Livorno - which carries on post war.

(Germany had great regional differences too and was similarly recent in terms of unification - but the comprehensive destruction of the opposition to fascist rule, particularly the far left, was of a far greater scope than happened in Italy).