Pius III died less than a month after his elevation. All that I can find, after browsing on Wikipedia, is that "this selection can be seen as a compromise between factions, Borgia and della Rovere, picking a frail cardinal with long experience in the Roman Curia over the kin of either Sixtus IV or Alexander VI."
That is all well and good, but there is a difference between picking a compromise candidate and one who, it seems, was literally on his deathbed before he was even elected, seeing as he couldn't attend the vote...
One of the things that is most important to understand about the Papacy during this time period is how political the position was. During the fifteenth and sixteenth centuries, both France and Spain (or the Holy Roman Empire under Spanish Hapsburg rule) vied for control of the Holy Office and dominance in the Roman Curia. The Papacy had been directly under the control of the French during the Avignon Exile, and when the Papacy returned to Rome France still held enormous influence through their cardinals and ambassadors present there. I don’t know all that much about French influence in Rome aside from what I have just mentioned, so the majority of my focus will turn towards Spain.
Spain’s influence in the Italian peninsula dates back to 1443, when Alfonso V of Aragon conquered the kingdom of Naples. Two years following the conquest, Alfonso de Borja, a Spaniard and the Bishop of Valencia, was elected pope and coronated as Pope Calixtus III. Calixus III disrupted the hold Italian and French popes had held on the papacy for centuries, using his newfound influence to increase the standing of fellow Spaniards within Rome. The Spanish influence in the Vatican was so great that after Calixus’ death in 1492 his nephew was elected as Pope Alexander VI. The Borja family would use their leverage with Spain and their power as popes to gain extreme power over some of the northern Italian city-states. Alexander VI began to consolidate power for his family in Florence and the Papal States, mixing his family eternally into Italian politics, Italianizing their name as Borgia, and dwindling the influence the Spanish kings had over their Spanish pope.
Here is where Cesare Borgia comes into play. Cesare was Alexander’s nephew, and the Captain-General of the Church. During the end of the fifteenth century, France and Aragon (Spain) were at war over the Kingdom of Naples. Alexander originally allied himself with France over his former Spanish allies. By his death, French and Spanish soldiers were camped all around Rome, with Cesare’s Papal army inside the city. The situation was, suffice to say, tense. France wanted Cardinal Georges d'Amboise to win the Papal election, and directed its cardinals to unify and vote for him. Cardinal d’Amboise is said to have bribed cardinals to vote for him, and was the favorite of the Roman citizens. Due to its conflicts with the late Alexander VI, Spain had lost any sway with the Italian cardinals and no Spanish cardinal stood a chance in the elections. Because of this, Spain sought to elect anyone that was not favored by the French, and Ferdinand II directed his cardinals to vote against d’Amboise and Cardinal Guiliano della Rovere, who was in favor with the Italians.
During the first ‘round’ of elections, della Rovere won 15 votes, d’Amboise won 13, and deacon Francesco Todeschini Piccolomini won 8. Piccolomini won the votes of the Spanish cardinals, who refused to vote for the other two French-backed candidates. In the second round, d’Amboise, having lost much of his initial support, threw his support behind Piccolomini. Piccolomini, too ill to even attend the vote, won the election due to the support of the Spanish cardinals as well as d’Amboise and his remaining supporters. Piccolomini would have to be ordained a bishop before he could be made pope, and this elongated process only wore down his health. He would die a mere five days after his coronation, to (ironically) be succeeded by della Rovere as Pope Julius II.
Sources:
Kamen, Henry. "Structure and Politics." In The Spanish Inquisition: A Historical Revision, London: Weidenfeld & Nicolson, 1997.
Pattenden, Miles. "A 'Spanish Avignon'?: The Spanish Faction and the Monarchy of Phillip II." In The Spanish Presence in Sixteenth-Century Italy, Edited by Piers Baker-Bates. Farnham: Taylor and Francis, 2015.