Looking only at the case of the defeat of the Inca Empire (as opposed to other societies like the Aztecs), the role of disease was that disease made the Inca Empire disunited at the time when Pizarro came to South America.
Not long before Pizarro's arrival, the Inca emperor and most of his court, along with his heir, all died of smallpox. When that happened, Atahuallpa and his half brother Huascar got into a fight over who should be emperor next. This civil war divided the empire and made it much weaker than it would have been if the disease had not hit.
Thus, disease helped to split the Inca Empire by killing the emperor and causing a civil war over who should be the next emperor.