why we can use the equipartition theorem for translational motion of molecules at room temperature and above because quantization is unimportant
What they mean is that for translational motion the quanta are so small that using classical mechanics produces no error, i.e $k_BT$ is far far bigger than the energy gaps between translational quanta, ($k_B$ is the Boltzmann constant) the same may be true of rotational motion where quanta may be only be separated by a few wavenumbers vs $k_BT\approx 210$ wavenumbers at room temperature. It is generally true for rotational motion but not all e.g. H$_2$. Vibrations have quanta of at least hundreds of wavenumbers  and $v=1 , v=2$ etc. are hardly populated at room temperature.
We can only use the equipartition theorem if $k_BT \gg \Delta E$ where $\Delta E$ s the gap between energy levels, i.e the fact that there are discrete levels is no longer important. This can occur if the energy gaps are v small or the temperature very high or both.
(Technically the 'equipartition theorem' of classical statistical mechanics states that the mean (average) value of each independent quadratic term in the energy is equal to $k_BT/2$)