First off, remember that many Roman historians, like Tacitus, were themselves of senatorial rank. They had a keen interest in anything that affected senators, such as the changes made by the Principate.

Romans hated kings. Before the Republic, Rome was ruled by seven legendary kings, starting with Romulus. Some of these kings had quite good personal reputations, but the institution of kingship was hated. The last king, Tarquinius Superbus (Tarquin the Proud), and his family behaved so badly that they were overthrown and the Republic instituted by Brutus. Tarquin's son raped the exceptionally virtuous Lucretia, who committed suicide after telling her father, husband and Brutus. Brutus and Collatinus (Lucretia's widower) became the first two consuls. This is all in the first book of Livy's Ab urbe condita, if you'd like the full story.

That's not to say that Romans of the Principate looked back fondly on Republican times because they saw themselves as living under a king (although that was something that seems to have been feared when Caesar was appointed dictator for life). It's more that they saw the Republic as a superior governmental system, especially morally. The distribution of power in the Republic was quite careful and, to them (that is, to the senators who wrote the history books from which we get this information), ideal. That distribution of power changed with the Principate.

The key point, I think, is that senators of the Principate felt that their own role within Rome had been curtailed. There was a sense that the cursus honorum (the order of public offices available for senators) had been hollowed out; the offices still existed, but the range, power and importance (especially of the consuls) decreased because so much lay in the power of the Princeps. Augustus was acutely aware of this problem, which is why he put so much effort into describing his creation of the Principate as a "Restoration of the Republic".

So the simplistic answer is that the people who write histories, whether now or then, are people for whom the republic would have been great. Mainly rich white dudes. There's something to be said for the idea that under the emperors (pax Romana and all that) average Romans (maybe not slaves or women but back then they were generally just screwed anyway, literally and figuratively) had a better quality of life. But whether you're a 2rd century historian with no power but a senatorial title or an 18th century aristocrat who thinks the king is getting too big for his boots, the Republic looks pretty good. I kind of agree with them at times--but it does help to create a unified and stable society when you are beating the shit out of opponents in war every summer, and that's just not sustainable in the long term and then shit gets crazy with civil wars.

That being said, there is a persistent tendency among the Roman elite to look back at previous eras with longing. This can be seen from a very early era. Cato the Elder, writing in the 2nd c. BC, so well before the transition away from the Republic, is already assailing the mores of the contemporary Romans. Read the beginning of his de agricultura for some great complaints about the new fangled professions of moneylending and trade. This continues all the way up through everyone, from Cicero and Cato the younger through Augustus, who has a massive program of 'renovatio' to institute a new age. Also lots of laws on morality that his female relations don't take too seriously. So whatever political things there were (both ancient and modern) that made people miss the Republic (and it's vital to remember that almost all historians were senatorial elites, wistful for a time when they actually had power), you also have to remember that the Romans loved nothing more than to talk about how much better everything used to be.