This is a non authoritative translation of a part of the law that I believe TFA missed, legal-Italian to plain-Italian to plain-English (as good as I can get it). Italics are mine.
Online advertising spaces and sponsored links in search engine result pages that can be viewed on the Italian territory during a visit to a web site or when using on online service on landline or mobile network, must be bought exclusively by companies with a registered Italian VAT account. This applies also to the case in which the sale has been made by the means of media centers, third parties and advertisers.
Think about the implications of the part in italics. Your US company buys an ad in English from Google aimed to the US market. Unfortunately I end up seeing it from my computer located in Italy. Ops, somebody is in trouble now, either you, Google, me or a combination of those three parties. There is nothing in the law about what happens in case of violations and to whom it happens.
Furthermore TFA missed that the law binds companies like Google to register a VAT account in Italy, not to pay taxes there. They'll end up paying just VAT there, which by the way comes from Italians, not from Google. The law aims at quantifying the turnover of those companies in Italy, which can only be estimated now. Unfortunately the way it's worded makes it difficult to enforce.
Luckily a motion (in Italian, Google translation to English here) has already been filed to suspend it. For another take on it you can read this Google translated post from wired.it.
PS: odd thing to do for me on Christmas morning :-)