The first thing that needs to be acknoledged is the Italy is by no means the only exotic setting for Shakespeare's work. He set plays in France and Egypt (following history), but also set non-strictly historical plays in Denmark, Greece, Scotland (which was then a foreign country), Bohemia (which he famously described as having "coasts") as well as a magical non-existant Mediterreanean Island.

He did set quite a fair number of plays in Italy however. There are a number of reasons for this, but what they ultimately boil down to was that Italy was a major centre of culture in Europe both during Shakespeare's time and for centuries prior. This meant that a lot of the books that were written in Europe, were written by Italians, especially fiction. Shakespeare drew on a lot of Italian sources to make his plays, and kept the setting for many of them in Italy as it was originally. Italy was seen as wealthy and decadent by contemporary English, giving an excuse to add spectacle to plays, especially using certain cultural traditions such as the Venician masked balls. This wealth also simply meant certain plots only made sense when set in Italy. Its hard to imagine the Merchant of Venice or Othello taking place in cities that were not centres of commercial wealth. In the same way it make sense to set Independence day is set in the U.S. and not Barabados because of relative military strength, it only made sense to set a lot of Shakepspeare plays in Italy due to comparative wealth.

If you want a lot of detail in an easy read I recommend this article by John Mullan, UCL Professor.

https://www.bl.uk/shakespeare/articles/shakespeare-and-italy

If anyone is wondering about the relative numbers, I've tallied his play's locations up. I haven’t listed off all the histories since they’re all set in England (although many do have scenes in France).

I’ve repeated plays if they happen in multiple locations.

England: 3 + 11 histories = 14 (Merry Wives of Windsor, King Lear†, Cymbeline†)

Italy: 9 (Two Gentlemen of Verona, Much Ado About Nothing†, The Merchant of Venice†, The Taming of the Shrew, All’s Well That Ends Well†, The Winter’s Tale†, Romeo and Juliet†, Othello†, Cymbeline)

Roman Empire: 4 (Coriolanus†, Julius Caesar†, Titus Andronicus†, Antony and Cleopatra†)

France: 2 (Plus some of the histories) (All’s Well That Ends Well†, As You Like It†)

Greece/Greek Colonies/Ancient Greek – related settings like Troy: 6 (The Comedy of Errors, whose source isn’t set in Ephesus, but is set in another ancient Greek city, Epidamnos, A Midsummer Night’s Dream, Pericles, Prince of Tyre†, Troilus and Cressida†, Timothy of Athens†, Two Noble Kinsmen†

Bohemia: 1 (The Winter’s Tale†)

Cyprus: 1 (Othello†)

Denmark: 1 (Hamlet†)

Illyria: 1 (Twelfth Night)

Navarre: 1 (Love’s Labour Lost, in modern-day France, I believe. Not based on a particular story, but it references historical figures)

Scotland: 1 (Macbeth†)

Vienna: 1 (Measure for Measure†, although the characters have Italian names, presumably because Shakespeare based it on an story by an Italian author)

A non-specific island, presumably Mediterranean, with allusions to the New World: 1 (The Tempest)

†My (rather surface level) research indicates these plays are likely based on a story / history set in that location. The extent to which they're adaption vs inspired might vary quite a bit.

The one piece of analysis I'll point out from doing that, is that most of his plays, whether they're set in England or abroad, are based on either published stories, or historical events, or both. So, while the question of why he chose to adapt so many stories set in Italy is a valid question, he didn't directly choose to set the stories there himself. The "choice" of setting is simply a result of the choice of story.

E: Added Two Noble Kinsmen which I swear I had down at some point. Also relevant to note that Twelfth Night has Italian names, is based on an Italian story and might well be set in Venetian territory, and that Cymbeline, while physically located in what is now England, is set in Roman times. Whether or not it's intended to be set during Roman rule of England is unclear to me - Wikipedia describes the character Cymbeline as a vassal king, but the historical king appears to have been independent. Either way, the Romans are heavily involved.

I'd like to ask a follow-up / piggy-back question:

What would a lay commoner of the Elizabethian era know about Italy, France, and other such nations? Was travel and/or first-hand accounts of foreign lands typical enough among commoners of this era that ordinary English citizens would know that Italy was a center of commerce and wealth in the time, or was that sort of knowledge really only typical among the nobility and groups (say, merchants and soldiers) that might have traveled abroad on official business?