The Field of Fire
The Gardener king of the Reach and the Lannister king of the Rock brought fifty-five thousand men against Aegon's ten. All three dragons flew that day. Four thousand men burned, House Gardener ended entirely, and Ser Harlan Tyrell ÔÇö the Gardeners' steward ÔÇö knelt and was handed Highgarden for it, which is the whole answer to why the Tyrells rule the Reach.
The fuller tale
The two richest kingdoms in Westeros combined against the invader and brought fifty-five thousand men to a field of dry summer grass in the Reach. King Mern IX Gardener came with his sons and grandsons; King Loren I Lannister came with the chivalry of the west. Aegon had about ten thousand, most of them men who had knelt within the last year and might not stay knelt.
All three dragons flew that day. It was the only time in the Conquest that Aegon, Visenya and Rhaenys took the field together, and they did it because the odds required it. The grass caught. Four thousand men burned and a thousand more died in the crush; the numbers matter less than the shape of the thing, which was an army that could not reach its enemy.
House Gardener ended there ÔÇö every male in the direct line, in one afternoon, the only one of the old royal houses to be wiped out root and branch. Loren Lannister broke free of the fire, looked at the arithmetic, and knelt while there was still a family left to kneel with. He kept Casterly Rock and lost a crown, which given how House Gardener's day had gone was an excellent trade.
How the Tyrells got Highgarden
With the Gardeners gone, Highgarden was standing open. Its steward, Harlen Tyrell, surrendered the castle without a fight and was given it, along with the Reach, the wardenship of the South, and a lordship over families that had been kings when his own were counting sacks of grain.
Three hundred years later the roses are one of the great powers of the realm and half the Reach still remembers exactly how they got there. Nothing in this history explains the Reach's internal politics better than the fact that its ruling house is descended from a servant who opened a gate.
Elsewhere in the wiki
 See the world as it stood in 1 BC What became of it  if an ending is written, it is here. Spoilers.
The chronicle records no ending here ÔÇö or none the maesters have set down.