It basically comes down to practicality. First, just to clear up, the process for electing a pope is effectively an election done by the College of Cardinals. A cheap and accessible book that'll give you a great deal of information on the electoral process and its history is Behind Locked Doors: A history of Papal Elections by Frederic Baumgartner.

So as the System goes, the College of Cardinals must convene and cast votes to select a pope -- the College of Cardinals, in fact, exists for the purpose of electing a pope. And the pope has almost always been from the College of Cardinals themselves (save for a handful of exceptions). So there is one huge practical limiter to having an effective college of Cardinals which have historical worked in the favor of the Italians -- geography.

The Church has been centered around Rome, and the Catholic population of the Rome had historically centered around Western Europe. So most of the priests and bishops would live in these areas and in turn most of the Cardinals. Further, Cardinals need to be in a place where they can easily and quickly get to Rome so that they can elect the pope when time comes. Additionally, since Popes chose Cardinals, and since they are still human beings with political leanings, they would often choose cardinals that came from their own background -- usually French and Italian for much of the Middle Ages. During the Renaissance, when Popes began to exclusively come from Italy, the position of Pope itself would often be the object of a great deal of political intrigue between the great dynasties and families of Italy at the time, and they would give cardinalships to people within their own noble families and allied ones (it's also important to note that Cardinals and Bishop were almost exclusively those of the nobility for much of history as well, and in turn would choose other nobility that would usually be confined to their own sphere of knowledge. So you don't get many Cardinals chosen from Far off in Sweden or the Netherlands). So the politics of the time, in combination with the geographic limitation would ensure a college of Cardinals that was almost entirely Italian (save for the few times in the middle ages when they would also be French and there would be bit of a power struggle there, see: Avignon Papacy).

Speaking of the Avignon Papacy, the ensuing Western Schism that would follow it would also encourage the Italian popes to follow to maintain the Papacy in Italy. You can see that in the earlier days of the Church when it was still not the official religion of Rome and there was still Christian unity with the East, there were several popes that came from the Syrian/Anatolian region, reflecting the demographic reality of the church and the relative geographic connectedness (though it is important to note that before the 13th century, Papal election was much less formal and much less ritualized and explicitly set, which led to a fair bit of anti-popery and such).

When did this all begin to change? Well, one of the big factors demographic change -- there is a significant catholic population in the US as we get into the 19th century and Irish immigration, an absolutely enormous Catholic population in South America, and a rapidly growing one in Africa. Now this wasn't reflected in the actual distribution of Cardinals and it still isn't. Europe as 54 cardinals and 1/4th of the Catholic population (albeit one that isn't very practicing), whereas South america has a slightly larger catholic population and only 11 cardinals (and one that is much more devout, as far as statistics on church attendance and belief in God go) than those in Europe. As travel became easier, especially with the advent of planes, there was a greater number of cardinals coming from a great deal more countries, especially outside of Europe. Additionally, just the fact that Italians compose such a small part of the Catholic population gave obvious practical motivations -- you can't justify the hegemony of Italian popes forever. Pope John Paul II was elected for a number of reasons, but the effects of Globalisation on the Catholic church and the expectations of its community, I would gander, played into it. Elections were happening on much less of a nationalistic line, and the increasing diversity of the college of cardinals also helped make this happen. The next two popes are outside of the 20 year rule, but i hope the mods won't mind me just making one quick mention of Pope Francis -- he in many ways shows the reality that the epicenter of the catholic church is much more Latin American, and there's a need to reflect this in the leadership.

Hope this helped!