The Celcius scale was originally based on the freezing and boiling points of water, so 0 °C was chosen as the freezing point until 1954. But now, the size of one degree on both Celcius and Kelvin scales is defined as 1/273.16 of the difference between absolute zero and the triple point of VSMOW (for various reasons relating to the actual measurement), but the 0 point of the Celcius scale was left as the freezing point of water which means that even though both scales are defined at 0 K and 273.16 K (the triple point), the Celcius scale still starts at the freezing point and 0.01 °C was defined as the triple point of VSMOW, which means 0 °C = 273.15 K.
Absolute zero is the coldest possible thermodynamic temperature, 0 K. The triple point is the temperature and pressure at which solid, liquid, and gas phases can exist in a stable equilibrium. If you look at the phase diagram below, it's the point at which the solid, liquid, and gas regions all meet. For water, the temperature of the triple point (273.16 K) is remarkably close to its freezing point at atmospheric pressure (273.15 K). Because 0 °C was originally defined as the freezing point of water, instead of changing the zero point when they redefined the scale based on absolute zero and the triple point, they decided to define the triple point as 0.01 °C.
We add 273.15 to °C to get kelvins because that's the difference between 0 K and 0 °C. We don't add 273.16 because the triple point is not 0 °C.