I am not understanding how can vapour coexist at temperature 273.16 K?
Molecules in a liquid don't all have the same energy. The energies of the molecules can vary from a finite minimum to a very large value. The average energy of the molecules gives us the temperature of the liquid. At a certain energy molecules will have enough energy to evaporate, even if the water temperature is 0 degrees celsius. Because of these two effects it is possible for the water to exist as solid, liquid and gas at the same time.