The economic growth, or at least the seeds thereof, largely came during Park Chung Hee's dictatorship, which ended in 1979 with his assassination. GDP per capita increased over tenfold during his tenure. The shift towards democracy afterwards was tumultuous for a while, with the first real civilian presidency being Kim Young-sam's in the 90s.
You're definitely right that the growth story is one of the most amazing in history. It's a country roughly the size of Indiana with no natural resources to speak of, and over 50 million people (over 10 million more than California, the most populous state) crammed into the few flatlands it has, being a very mountainous country. Yet it was able to go from an agrarian society to a highly developed nation, with the 12th largest GDP, home to feats like building a third of the world's tonnage of ships. It's absolutely wild.