The fall as a whole is the result of a whole lot of mismanagement of state affairs for quite a long time. I'll leave out the Autumn War (when Ala Mhigo tried to annex the Black Shroud a hundred years ago, sparking a war with Gridania that led Ishgard to unite with Ul'dah and Limsa Lominsa to come to Gridania's aid). Instead, let's focus specifically on the Ala Mhigan royal family and the monkhood. The interactions of these two groups are what eventually caused the downfall of the city.
The tribes throughout Gyr Abania were once small, warlike, and constantly at each others' throats, but the Highlanders were able to bring them together as one and raise the city of Ala Mhigo (some time between 1,000 and 1,300 Sixth Astral Era). The idea of martial arts as sacred persisted even though the age of constant battle was over, and thus rose the Fists of Rhalgr, the monk order, which teaches that one can become close to the god of destruction through strict discipline that empowers them by channeling their "life force" (aether) through "chakras".
This order eventually became religion, with the war priests claiming that their chosen battles were divine will. Rather than fight the movement, the Ala Mhigan royal family incorporated them into their army, giving them considerable power, protected status, and political sway. By the twilight of the Sixth Astral Era, as both a political force and the official state religion, the Fists of Rhalgr had control over a great deal of daily life, which deeply unsettled the power-hungry (and soon to be final) king of Ala Mhigo.
King Theodoric wasn't very into political finesse; he refused to allow the monkhood to contradict his designs in any fashion, even if it meant destabilizing the city. For instance, the monks claimed divine righteousness, so Theodoric claimed the divine right of kings, instilled in him by Rhalgr's wife (Nymeia). By this right, he banned the worship of Rhalgr in favor of Nymeia and tried to establish himself as the uncontested dictator of Ala Mhigan affairs. This sudden, drastic move away from Rhalgr was designed to incite rebellion by the monkhood, giving him pretext to imprison its leadership and extract coerced confessions of their plans to topple the royal family.
This concluded with Theodoric marching on the Fists of Rhalgr's main temple in 1552 and burning it (and everyone associated with it, from monks to their families to refugees seeking asylum) to the ground. Most knowledge of the monkhood vanished at this point, and afterwards Theodoric was known as the King of Ruin, a ruthless despot.
True to his name and aforementioned lack of political finesse, he slaughtered any royals towards whom he harbored doubts, as well as anyone who might have some tacit claim to the throne. After that, he moved on to influential nobles and merchants. So hopeless was it that Theodoric would stop this progression that even his own wife turned against him, conspiring to have him murdered. One of the conspirators, however, panicked and confessed to Theodoric, thus getting them all executed. The king withdrew, paranoid, and the public killings grew more frequent.
With people afraid to even leave their homes, the idea of revolution spread like wildfire - rising from whispers to a full on revolt that stormed the royal palace; the guards did not attempt to bar their passage, and Theodoric took his own life as his enemies closed in. This was no revolution of the people, however. The intelligence services of the Garlean Empire's XIVth legion, under the direct command of Gaius van Baelsar, were responsible. (While unconfirmed, wouldn't it be wild if the Garlean agents were also responsible for exacerbating Theodoric's behavior? Talk about a potentially well-played long-con from Gaius in his prime, before he lost his pride, honor, and marbles.)
In the aftermath of the king's fall, the Garleans stormed the city before any defense could be mounted, taking it in what might be called a bloodless and subversive coup if not for the fact that the bloodshed came after. The city was sacked - biblical-era levels of sacked. Magitek crushed what rebellion did not surrender or flee deeper into Eorzea, and the city was converted into a walled fortress to act as the Garlean foothold in the realm. The atrocities committed here (especially by his own creations) were a big part of what inspired Cid Garlond to defect to Eorzea to help it defend itself.
For the sake of completeness, from here, it took Baelsar five years to properly fortify Ala Mhigo and prepare for the conquest of Eorzea, but the 1562 assault on Mor Dhona, an attempt to take control of the aetherial streams before primals could be summoned against them as they had in Othard, went (as I'm sure you know) very, very badly.