ACuriousMind is right. Assuming the system contains a dissipative component, for any fixed driving frequency, when the rate of energy input from the driving source equals the rate at which energy is lost from any dissipative elements in the system, a dynamic equilibrium is reached in the system. At dynamic equilibrium the sum of the average potential and average kinetic energy are a constant.
What Crawford is trying to illustrate is what particular form of energy is dominant below and above the natural (resonant) frequency of the system. Below the resonant frequency kinetic energy is dominant and above the resonant frequency potential energy is dominant. And at the resonant frequency each form is equal.
Note that prefaced "For any fixed driving frequency ..." above. The total energy at dynamic equilibrium is a constant, but only at a fixed frequency. The total energy is a maximum at the resonant frequency. At the resonant frequency, the system has its maximum capacity for energy.
Now if there is no energy dissipation or the rate of dissipation is less than the rate of energy input - there is danger of the system 'blowing up'. But in real physical systems there is always nonlinearity. The nonlinearity may provide a safe release for energy once some limiting factor is reached, or in other cases (like the Tacoma Narrows Bridge) reach a point of disaster.
Keep in mind that the bold text isn't a derivation, its a way to qualitatively understand, so it's going to be very very unconvincing. Instead of imaging that you are trying to compute a number imagine you are trying to estimate it by a factor of 10 or 100 or even a factor of 1000 or more. That's how unconvincing it will be.
So. At equilibrium amplitudes, power supplied equals power lost to friction. How much energy is lost to friction? Well, when no power is supplied it basically dissipates almost all of the available energy $E$ in time $\tau$. So friction saps away something like $E/\tau$ as a power lost to friction. So the power supplied is something like $E/\tau$ too.
The conclusion is that we expect the power supplied and the power lost to be somewhat equal to the energy stored divided by $\tau.$
But this is unconvincing because the power lost to friction changed and wasn't constant, if we waited longer then closer to 100% of the energy would be dissipated but the average power would be much smaller because we included more time of lower power loss.
The whole thing is incredibly wild estimates, not much different than dimensional analysis.
Most of the energy $E$ is lost by $\tau$ if the driving force is absent. Okay. But how is it related to the stored energy? Even if the driving force is present, it will have to supply $E/\tau$ so as to compensate the loss. How can it be stored, then?? it is just nullifying the dissipation; it is not stored?
The energy stored is defined to be $E$. Because the letter $E$ is the symbol I made up for the total stored energy. And $\tau$ is an almost arbitrary time related to how long it takes for the friction force to dissipate almost all of the energy $E$. Why is it that almost all of the energy is what is dissipated? Because we choose the zero of energy to be located at that place the friction approaches. Its not the absolute zero. There is more energy there, there is, force instance rest energy associated with every particle. There is chemical binding energy associated with each electron being in some atom/molecule. There is some energy associated with the thermal motion of the system. You could extract that energy with antimatter, chemical reactions with more reactive elements, and thermal contact with colder objects. But there isn't any more macroscopic kinetic energy or potential energy available besides $E$ that's all that is available. And that's the state the friction drives the system to. And it never gets perfectly there but it gets close in time $\tau$ and so in time $\tau$ almost $E$ is dissipated. Because of the definition of the two symbols.
Edit At resonance the amplitude can grow. As it grows the friction increases. When the amplitude grows it grows and grows and grows until the amplitude is so large that the friction dissipated is now exactly equal to the driving power. And that friction power is somewhat approximately equal to $E/\tau$ where $E$ is the total available energy stored.
At another frequency the amplitude doesn't grow so it doesn't get up to a nonzero steady state. In a complicated and real system there might be many modes that can be resonant and there might be shifting of modes over time.
Best Answer
This is just a footnote to Name's answer (which you should accept because it's correct) to give a slightly more intuition based argument.
If the driving force changes slowly compared to the natural frequency of the system then the system can move fast enough to stay in phase with the driving force. So most of the time the system is already moving in the direction the driving force is pushing it, and the force will accelerate the motion so the resulting amplitude of the oscillation will be big.
If the frequency of the driving force is a lot higher than the natural frequency of the system then the system cannot move fast enough to stay in phase with the driving force. This means some of the time the driving force is acting opposite to direction the system is moving, so it's slowing the motion not accelerating it. This means the amplitude of the motion will be reduced.