Mainly three factors should be considered:
1) especially in Lombardy, a lot of people kept on reuniting and considering the jobs marked as essentials, in practice nearly half of the people living there kept a "normal life" -> go to job, stay in close contact etc;
2) it seems like, especially in Lombardy which is responsible for 2/3 of daily cases, that there's something wrong in the hospitals, I mean: it seems like that it is spreading through the hospitals, as it started in practice;
3) as I wrote before on the forum, on 9th March restrictions started to happen and in that day (and the next weekend) a lot of people travelled from the North to the South of Italy. Considering a median value of incubation time of 14 days, plus 3-4 days for relatively severe symptoms to appear, you reach more or less 26th March and I already wrote that the end of this week will make a very important point. Consider, in addition, that a lot of people making this travel were not exactly really smart, so some of them kept on going for shopping, pharmacies and it was estimated that 15% of people moving could have potentially had at least 1 symptom.