This question gets asked a lot, but the answer seems to often appear in small chunks.

I could find one answer on this topic here from /u/aquamarinade, though admittedly it covers it from the perspective of the Pope themselves.

Then there's this answer from /u/RomanItalianEuropean which is in answer to a different question but sort of incidentally addresses your question in the opening two paragraphs.

Perhaps the most comprehensive answer which I could find comes from /u/klesk_vs_xaero, who explores the relationship between lay Catholics and the Papal States during the Risorgimento here.

The Italian governmemt was very (very) cautious about the whole operation. Italians wanted Rome to be capital of Italy probably more than everything, as it it made perfect sense historically and geographically, but only the left-wingers (primarily the "red shirts" of Garibaldi) wanted to wage war against the Pope. And most italians rejected the scenario of a war not only out of respect for their Church, but also out of fear of retaliation by catholic powers; the powerful Napoleon III being the chief protector of the Pope.

When, on 17 march 1861, Victor Emmanuel II assumed the title of King of Italy, Rome (and the surrounding region of Lazio) was still part of the Papal Stats. The prime minister of Italy, Camillo Benso of Cavour, had the first Italian parliament vote the following resolution:

1)Italians unofficially "acclaim" Rome as capital of Italy (a rather curious formula, Turin remaining officially the capital).

2)Italy wants to make Rome the official capital via agreement with the Pope and Napoleon III.

Diplomacy, not war. In other words, Cavour wanted the Pope to cede Rome in exchange of money, protection, and privileges. Among those privilges, there was already the idea of leaving Vatican Hill to the Pope as a mini-state. Cavour called his project: "free church in free state".

In line with this policy, the Italian government even intervened to prevent attempts by volunteers led by Garibaldi to take Rome by force in 1862. However, the Pope wanted to keep Rome at all costs and denied the offer to come to terms. He felt secure, since Napoleon III supported him. Another attempt made by italian volunteers to take Rome was in fact stopped by Napoleon III in 1867 . When Napoleon III fell in 1870, Italy again asked the Pope, now without a protector, to come to terms. Again, a no. What to do now? The issue was debated at the Council of Ministers. It was agreed that Italian troops would seize Rome with a quick operation. The quickness of the operation was key to the success, Italians had to wake up with Rome capital of Italy and the King already proposing concessions. Rome was taken, Vatican hill was not occupied, the Pope was offered compensations. Then, certain laws giving privileges to the Pope and the Church in Italy were passed. It was the Pope who refused and called himself a prisoner of the new Italian state (until Mussolini and the Lateran treaties in 1929, but that's another story).

"Giovanni Lanza" dizionario biografico degli Italiani, Treccani,

Indro Montanelli, "storia d"Italia", l'Italia dei Notabili",