It appears the event is central to the Dragonlance setting as it informs the calendar; early years appear annotated “pre-Cataclysm” (PC). Example year 5000PC.
What is the cataclysm, how did it come to be?
The Cataclysm was the gods punishing all of Istar for the Kingpriest's hubris, but it was also an event that permanently changed the geography of Krynn. According to Shadow of the Dragon Queen (p6, The Cataclysm):
A mountain of fire fell from the sky, destroying Istar. The Blood Sea - a new ocean that split eastern Ansalon - consumed the empire. Coastlines shifted all over the world, sundering nations, drowning whole regions, and stranding ports miles from the sea.
You may want to read Tales: Volume II. The synopsis for that book on goodreads is:
The Kingpriest's arrogance brings the wrath of the gods upon Krynn. The result is the Cataclysm -- chaos and anarchy, despair and villainy...
Fandom isn't necesarily the most reliable source, but it has this to say:
The Cataclysm (Great Destruction) was a catastrophic event that occurred when the Kingpriest of Istar demanded to be given godlike powers in order to eliminate evil from the world. The gods threw down a fiery mountain upon the Kingpriest and all of Istar for his impunity.
According to dragonlancenexus:
It is widely believed that the Cataclysm occurred as a result of the Kingpriest’s overweening ambition. The Kingpriest sought godhood to not only purge the lands of evil, but of anything that did not agree with his viewpoints.
In the original Dragonlance Chronicles and Dragonlance Legends, the story/legend of the Cataclysm is mentioned a few times, most notably in connection with the Blood Sea. According to the created history, after the third dragon war, the Kingpriest of Istar angered the gods, who cast down a fiery mountain to destroy the city of Istar, as the continent was sundered, the sea rushed in, forming the Maelstrom and the Blood Sea. Legend has it that the sea is red because of the blood of the dead of Istar. Many kingdoms were destroyed or drastically altered by the upheaval.
I no longer have my copies, but that was the gist of it as I remember it.
I'm afraid my memory of the short story collections where they go into more detail is rather more shaky, but as far as I can remember, The Kingpriest was a follower of Paladine who believed the concept of balance was forced upon Paladine when he was weak. He didn't like it and wanted to destroy the religions/followers of the evil gods, as well as evil itself. Eventually he and his followers started killing/destroying anyone who didn't agree with him, including followers of neutral and even good gods. His hubris was eventually so great that he demanded that the gods heard his demands to make him a god and give him the power to eradicate evil in the world. The gods gave all sorts of warnings (either 13 warnings or 13 days of warning) and eventually destroyed the Kingpriest and the city by chucking a burning mountain/meteor at it. There followed all kinds of disasters, there was aperiod of darkness and the gods stopped answering prayers for a while.