Elections in 2018 gave a somewhat stuck parliament, with m5s (generic populists) holding 36% of seats, Lega (northern rightists/hard populists) 20% , PD (leftists) 18% and FI (Berlusconi) 16%. All parties have their internal problems, m5s and Lega are mainly a bunch of idiots, PDs are divided in incontrollable factions, Berlusconi is getting old. The government was set up by an alliance between m5s and Lega.
European elections in 2019 gave a completely different result, Lega 34%, Pd 23%, m5s 17%, FI 9%. In other words, the popularity indexes of m5s and Lega are inverted and this time Lega is much bigger than m5s. This is valid only for seats in the European parliament, but it is a big and official poll about how Italians think. So Lega has the interest to trash the present government, have the Italian parliament dissolved, run again the national elections, and with a bit of luck they could set up their own government (if you manage to get 40% you get the majority of parliament seats). Lega would like to get out of the euro system in order to be free to increase public spending, get rid of immigrants and generally speaking they are quite on the rightish side. So the Lega head asked for the government to resign.
What he had not realized, his votes are only virtual: he has not the majority in the Italian parliament. As soon as he asked for resignation, PD proposed an alliance to m5s that would lock Lega out of power. So yesterday the Lega head was deeply humiliated by the resignation speech of the head of government, that accused him openly of doing his own interests instead of the interest of the people. Now the Italian president has to find another head of government able to find a majority in the houses. The most probable outcome is a m5s-PD government, looking unlikely on paper, but it would be in their common interest to get some time and not to run the elections again. Even a "national unity" government m5s-PD-FI could be possible - it looks even more unlikely on paper, but who cares. If it turns out to be impossible to form a new government, the president will dissolve the houses and new elections will be held.
No immediate consequences to travelers till the end of this year. If election are held and the Lega wins, they could try their economic ideas and get the country into some chaos, but it is not as easy as it sounds. If a government can be formed, they first problem is that they have to find some money in the state budget to prevent a raise in VAT rates that would be automatic on January 1st, driving up all prices.