The origins of the state would end its life as the Austro-Hungarian Empire in 1918 began in 1515, when the King of Hungary Vladislaus II had his sole son and heir, the future King Louis II marry Mary of Austria, the granddaughter of Holy Roman Emperor Maximilian I and sister of future Emperor Ferdinand I. Ferdinand himself would also marry Louis II's sister, Anne as part of a wider mutual succession agreement between the House of Habsburg, which ruled the Archduchy of Austria and held the Holy Roman imperial crown, and the House of Jagiellon, of which the soon to be King Louis II ruled the Kingdom of Bohemia and Kingdom of Hungary. Fast forward to 1526 and King Louis II leads the army of Hungary into battle against the Ottoman Empire, who after taking Belgrade in 1521 were well posed to strike into the heart of Hungary. The resulting Battle of Mohács is an utter disaster for Hungary. King Louis II was killed and had no heir, and in the subsequent chaos, the kingdom fractured. At the time, the "King" of Hungary was an elected position, with the nobility of Hungary having the right to name their King. With Louis II dead with no heir, the nobility split and proclaimed two different kings. In the northern and western portions of Hungary and the Kingdom of Croatia, the nobility proclaimed Louis II's brother-in-law, Archduke Ferdinand I of Austria as King of Hungary while in the east, much of the lesser nobility and gentry proclaimed John Zápolya as King of Hungary. Ferdinand and John would dispute the throne of Hungary for decades. At one point, in the Treaty of Nagyvárad, Ferdinand I acknowledged John as "King of Hungary" while John recognised Ferdinand's rule in the north and west while also allowing for Ferdinand to become King of Hungary on John's death. Yet within a few years, John would have a son, John Sigismund Zápolya who would then be crowned King John II of Hungary in spite of the previous agreement with Archduke Ferdinand. Eventually, the Ottomans would invade Hungary again on the pretext of supporting John Sigismund and in a series of wars and conflicts between the Habsburgs and the Ottomans, the end result was that by 1540 Hungary was divided in three. The Habsburgs retained their position in norther and western Hungary along with Croatia, which would become "Royal Hungary." The central Hungarian lands around Buda would be annexed by the Ottoman Empire while John Sigismund and his successors would become Princes of Transylvania, which solidified into the Principality of Transylvania by 1570 as an Ottoman vassal. While all this may be confusing, this map should help the illustrate the division of the lands. Hungary would become something of a borderland for both the Habsburgs and the Ottomans, with an ill-defined border and near constant raiding by regular and irregular forces across the boundary. The famous Habsburg Military Frontier across Croatia and Hungary was an example of this.

The division of Hungary would come to an end in the latter portions of the 17th century. In 1683, the Ottomans advanced and laid siege to Vienna in what would be the opening stages of the Great Turkish War that would last until 1699. By the end, the Ottomans had been mostly expelled from Hungary, and the Treaty of Karlowitz would confirm this, with much of Ottoman Hungary ceded to the Habsburgs and the quasi-independent Transylvania also brought under Habsburg domination. The Habsburgs would hold the "Kingdom of Hungary" as a core part of their possessions until the final dissolution of their Empire in 1918. While this is straying outside of my topic area, internal pressures within the Austrian Habsburg state, including the Hungarian Revolution of 1848 would eventually lead to the reorganisation of what had been the Austrian Empire into the Austro-Hungarian dual-monarchy in 1867. The Austro-Hungarian Empire would endure until its collapse and subsequent partitioning in 1918 at the end of the First World War.

To summarise, the merger of Austria and Hungary into a single state is a story with roots in the expansion of the Ottoman Empire and the wars waged between them and House of Habsburg. Hungary was partitioned between the Habsburgs, Ottomans, and Transylvania, prior to the Habsburgs conquering the other two portions and reforming the Kingdom of Hungary as part of a growing Austrian Habsburg Monarchy.

I hope this helps answer your question and please feel free to ask any follow ups!