The Titans are, if you strip all the cosmic power and grandeur away, meddlers. They go from world to world and change those worlds to their liking, whether or not the current inhabitants want them to, and Azeroth is no exception. When the Titans arrived, they found the world a seething battleground for raging elementals who fought for the amusement of ancient unfathomable entities called today by the sobriquet 'The Old Gods'. Entities like C'thun and Yogg-Saron are an example of the true lords of Azeroth. The quoted text at the bottom of this page gives you an idea of how the Titans wrested control of Azeroth away from them and sealed them in prisons beneath the planet's surface. With the elementals banished to other planes and the Old Gods trapped, the Titans then rearranged Azeroth to their liking.
However, since the Titans re-ordering of the world is imposed, they require shepherds to maintain it. While the Titanic Watchers of Ulduar were set to maintain Yogg-Saron's prison, they were not entrusted with the maintenance of the world itself, possibly because they were not of it. From Galakrond's offspring, known today as proto-dragons, the Titans shaped flights of dragons to the vital (to them, at least) task of preserving their hard-fought order. Black, Blue, Green, and Bronze were each given dominion over a key aspect of the Titan's new world. And to Alexstrasza, the Red Dragonqueen, was given control over Life itself. This was the birth of the Red Dragonflight, perhaps the most beneficent grounp of enormous fire breathing reptiles one could hope to encounter. They are dragons, however, so don't take them for granted.
The five dragonflights had a balance established over the course of millennia, each working to sustain and direct its purview (so the servants of Neltharion, the Earth Warder, would concern themselves with things like earthquakes and volcanoes, while Nozdormu's bronzes were rarely seen since they were more concerned with time, and Ysera's greens with dreams) which meant that the red dragonflight was one of the more active ones, since its purview was life itself and there were a great many living things to be concerned with. In addition, they often worked with other dragons when those dragons' sphere of control coincided. If, as an example, a great many living things suddenly died from a huge explosion of magical power, this would concern Malygos and Alexstrasza for different reasons, and thus their flights.
There was established under the great Wyrmrest Temple (which itself existed at the end of the so-called Path of the Titans that descends from Ulduar in the Storm Peaks and presumably once led further south down the united continent of Kalimdor before the Sundering) a vast chamber where the five dragon aspects could come together and meet. For thousands of years, the dragons continued in this way, each keeping to its own duties. What happened to change it?
The short story is, they failed. Perhaps their elevation by the Titans blinded them to the smaller, more fragile, shorter lived inhabitants of Azeroth and kept them from paying attention. While great empires rose and fought terrible battles, the dragons remained aloof, keeping to their ancient duties and rarely interfering even as the trolls raised their first, greatest empire and led it to war with the aqir. When the trolls and aqir smashed each others' nations into ruin, Azj'Aqir splintering north and south and eventually becoming the nerubians and silithid and the trolls beginning their slow decline, the dragons seem to have taken little to no notice of them.
Then someone... possibly a troll, possibly some other race... found the Well of Eternity.
We're still not really clear as to why the Well existed. Some sources claim that the Titans carefully shaped it in the center of the great single continent of Kalimdor, a vast font of pure magical power that drew energy not from the Twisted Nether but instead from the Great Dark Beyond between worlds. Thus, it was a pure force of magic without any of the associations of other kinds of magical energy such as fel power (the deformative magics of the demons) or even elemental or spiritual power granted by other entities. Why the Titans would make such a thing is unknown. Why the dragonflights simply allowed the mortal races to make use of it is even more unknown. We do know that a tribe of night dwelling humanoids found it, learned to use it, and in so doing became the Night Elves as we know them.
From this simple tribe (perhaps a lost group of trolls from the destroyed Gurubashi or Amani empires following their battles with Azj'Aqir, perhaps not) rose the towering civilization of the Kaldorei, with its vast and towering city of Zin'Ashari directly on the shores of the Well itself. The trolls were pushed back and defeated, driven to the corners of the world the elves didn't want badly enough to justify the cost of pushing them back further. And the dragons did nothing. While the red dragons were the guardians of life, they took no action, perhaps believing that this was all part of the natural course of life. One nation rises, another falls, just as how beasts contend in the world.
Whatever their reasoning, it's clear that the five major dragonflights were extremely hands off in their approach up to this point. Consider the events of the Great Sundering: a major malevolent entity from beyond the world (the Dark Titan Sargeras) used his power to dominate the minds of or seduce the night elves' greatest mages and leaders. An orc, dragon and human travel back in time and change the course of history (although to be fair they don't accomplish all that much, as history still unfolds more or less as it would have without them). You'd think these would be of immediate concern to at least three of the dragonflights, and yet when they finally take action it's too little and far too late.
To be fair, though, this isn't entirely the fault of the four aspects who were still sane by this point.