I do not want to see Korean National Holidays or my friends' birthdays on my KakaoTalk Calendar.
On the left side of KakaoTalk Calendar you can choose to view/hide "Holidays in Republic of Korea" and "Friends with Birthdays" by clicking the check box.

If you don't see the menu, your side menu might be set as hidden. In this case, open the menu by clicking the three-lined icon on the top-left corner.