After the first case, Italy did not take the disease very seriously. Schools were closed, but people thought this closure was "a holiday" and transitions between cities and travels increased. As a result of this mobility the disease has increased the spread throughout Italy.

Italy has a feature like this, it is the second country in the world with the highest old population after Japan. Since the disease also had the fatal effect on the elderly, death rates started to increase suddenly. The average age of people who died is 79, 4.
Another factor was the delay of the closing of borders and public places. Tourists from all over the world travelled to Italy, visited the museums and public places. Who is infected by the disease, who is not infected is unknown. So coronavırus found another area to spread itself.
One reason why there are so many cases and fatalities in Italy is the delay in testing. The government waited for the symptoms to appear for testing. However, in some cases, the disease showed no symptoms and passed on to another patient, and did not harm the carrier. The government imposed a curfew but the Italians did not take it too seriously, they continued to do their daily work again. Another reason is the the Italian health system is bad. Italy has the lowest number of the hospital beds in Europe.
As a result all these caused the increase of the pandemic. And we will see what will happen the next days.