Aegon lands at the mouth of the Blackwater
Aegon Targaryen came ashore with his two sisters, three dragons, and an army small enough to be a joke, and raised a wooden motte on a hill where a city would grow. He had a map of Westeros as one kingdom and no legal claim to any of it ÔÇö which turned out to matter less than Balerion did.
The fuller tale
For a hundred years House Targaryen had sat on Dragonstone being a minor power with a major asset: lords of a rock, married to the Free Cities, occasionally hired to settle other people's quarrels in the narrow sea. Then Aegon and his sisters landed at the mouth of the Blackwater Rush with three dragons and an army small enough that the local lords rode out to look at it with more curiosity than fear.
The stated cause was an insult ÔÇö Argilac the Arrogant, Storm King, had answered Aegon's offer of alliance and marriage by cutting the hands off the envoy and inviting him to come and take them. The real cause was a painted table. Aegon had a map of Westeros carved and painted as one country, in one piece, and had been looking at it for years. Nobody had drawn Westeros that way before. Everything that follows is the map imposing itself on the ground.
He raised a wooden palisade on a hill above the river, called it Aegonfort, and had himself crowned there by a septon in front of a few hundred people, which at the time was funny. Within two years it was not.
Three dragons and no claim
Aegon had no legal right to anything west of Dragonstone and never pretended otherwise. What he had was Balerion the Black Dread, Vhagar under Visenya and Meraxes under Rhaenys, and a shrewd sense of when not to use them. The Conquest is usually remembered as fire, but the great majority of it was lords doing arithmetic and kneeling: seven kingdoms, and only three real battles.
Where the fire did come, it was used to make an argument that could not be answered ÔÇö and to make it once, loudly, so it would not have to be made again.
Elsewhere in the wiki
 See the world as it stood in 2 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.