In the previous piece each component of the velocity came out with a mean of \dfrac{1}{2}kT. Seeing why that form appeared widens enormously the range of things it applies to.
The key is that the energy had the form \dfrac{1}{2}mv_x^2: proportional to the square of a variable. We shall call a variable of that shape a quadratic degree of freedom.
The figure is a mass on a spring. Its potential energy is \dfrac{1}{2}aq^2, a square again, and its kinetic energy is \dfrac{1}{2}mv^2. So this system carries two quadratic degrees of freedom.
Work out the mean energy of a quadratic degree of freedom and it comes to \dfrac{1}{2}kT, whatever the coefficient. This is the equipartition theorem.
The two panels in the figure are wells of different depth. Take the left as a stiff spring and the right as a soft one. The soft one spreads far wider, and by just so much its energy at any given displacement is smaller. The upshot is that the mean energy, shown as the shaded area, is the same in both.
The reason lies in the shape of the integral. The coefficient can be absorbed by rescaling the variable, so it never survives into the average. Stiff spring or soft, heavy molecule or light, so long as it is a square it is \dfrac{1}{2}kT.
That being so, the energy of a system follows from nothing but counting how many quadratic degrees of freedom it has.
The left of the figure is a monatomic molecule. It can move in three directions and no more, giving three kinetic terms, \dfrac{3}{2}kT in all. That agrees with the previous piece.
The right is a diatomic molecule. Besides the three translations there are two rotations that turn the axis (rotation about the axis itself is not counted), and a vibration in which the two atoms stretch and compress. Vibration enters as a square in both position and velocity, so one vibration counts as two. Seven in all, that is \dfrac{7}{2}kT.
The energy goes as the temperature, so the heat capacity is the constant of that proportion. With f degrees of freedom it is \dfrac{f}{2}k per molecule, or \dfrac{f}{2}R per mole.
The figure sets those out. \dfrac{3}{2}R for a monatomic gas, \dfrac{5}{2}R once rotation joins, and \dfrac{7}{2}R with vibration as well. The heat capacity can only take discrete values.
And indeed the heat capacity of monatomic helium or argon is \dfrac{3}{2}R, and nitrogen or oxygen at room temperature is \dfrac{5}{2}R. The agreement is good.
And yet a diatomic molecule at room temperature came to \dfrac{5}{2}R. The two from vibration are not in the count. Cool it further and the rotational share goes too, leaving \dfrac{3}{2}R.
The staircase in the figure is that. From cold upwards: translation alone, then rotation joins, then vibration. If equipartition held as stated there would be no staircase at all, and the top step would run at every temperature.
It collapses because energy can only take discrete values. Where kT is smaller than the spacing of the steps, that degree of freedom is stuck on the bottom step. This is called freezing out. Equipartition was only ever a statement for kT large enough. The next piece takes this freezing out head on.