To begin with, Vatican City is not necessarily synonymous with the Holy See; one is a sovereign state, the other represents the jurisdiction of the bishop of Rome. That palace's owned by the latter and is located on Italian soil.

As for the reasons as to why, shortly before the Lateran Treaties were to be signed it was agreed by both parties that the new state had to take as less space as possible so that the country in q. could be administered more efficiently...

...actually the first drafts of the Treaties included that palace within the borders of the soon-to-be Vatican City; however, Pius XI saw fit to exclude it - along with a handful of other properties located throughout Rome - from becoming part of the country proper. Indeed, during the preliminary talks the Pope appeared prone to making large concessions to the other party and therefore his rather accommodating disposition must've had quite an impact!

On the other hand Victor Emmanuel III (the then-King of Italy) was unwilling to cede more than what was strictly necessary so there's also that to consider; without any doubt his subborness, coupled with the Pope's flexible stance, contributed greatly to such a curious outcome. BTW my source is this article on the Osservatore Romano.