In large part it was due to the Emperor himself. He had grown old, and as he did so he increasingly acted less and less an emperor and more and more a leading citizen. He spent less time on civil duties and more time on charities. Although he never ceased to attend meetings or perform his official duties, he was clearly disenchanted with his role. He also had a period where he was further sidelined by health issues. During part of that his daughter Isabel acted as regent for him. So the position of emperor was increasingly seen as unnecessary. The government was running pretty well without an active emperor.

At the same time he didn't appear to have an appropriate heir. His sons had died. His daughter didn't seem to want the crown, and he didn't want her to have it; evidently he preferred a male heir to be found. It probably didn't help that the public disliked her husband. Her sons were also not preferred as they were not in the direct male Braganza line. There were two men who were, but they were excluded for being illegitimate, and not a direct descendant of Pedro I respectively. So when Pedro II died, there was going to be a messy succession. There simply wasn't a single heir that would be well accepted by the family, the political class, and the public at large.

Brazil also had by that point developed a pretty robust democracy. Its parliament was active with multiple parties. There was also a growing movement of republicanism. It wasn't a popular movement. The public as a whole loved the monarchy. But it was supported by the military and the intellectuals. And in the 1880s it picked up the plantation owners. They had seen their slaves freed without any kind of compensation. They blamed the emperor, and especially Isabel who was acting as regent when the emancipation law was passed. The emperor as well seemed pretty happy with the idea of a fully republican government and there was discussion going on in the palace about how to make such a transition.

This is where the coup plotters come in. All of this wasn't necessarily known to the public. But to anyone with access to the palace it was clear that a change in government and likely in constitution was coming. So they moved into action. It took little prompting to get the Emperor to accede to the coup, and the public at large was presented with a fait accompli. And frankly the coup was presented as a transition to a fully republican government. That it would become a military dictatorship wasn't clear at the time.

Had the Emperor wanted he probably could have resisted. He had the popular opinion on his side. He also had plenty of supporters in the military. But he was already working on abolishing the monarchy, so why not accept it? Likewise he had no strong well established heir to take up the banner. Isabel didn't have her father's support, and evidently didn't really want the throne anyway. Without a monarch the monarchists didn't have a leg to stand on. Had there been a Pedro III waiting in the wings, it's unlikely that a coup would have even been tried. It certainly would have had less chance of success. But as things were, there wasn't much of a viable alternative.