To Koreans, Independence from Japan was not directly mean that Korea could have their own government. Different "big" countries decided by themselves that Korea is unable to choose what's right and wrong by themselves thus they will take care of us. That was what they "claimed". So, they just separated Korea into two and the upper part was controlled by USSR while the lower part was controlled by the USA. What do you expect two parts of Korea to become? They follow the different idiology. So the difference in idiology caused two parts to have conflict and there it is, Korean war. Then? The country which used to be one becomes two. The People's Republic of Korea (North Korea) and the Republic of Korea (South Korea).
When we say the day that Korea's government has been established, it's the day that South Korea got it's own government.