What is meant by the signal energy to alias ratio and how is it quantified?
You add the energies in the signal and aliased curves.

Now if you assume the aliasing is completely de-correlated with the signal, you can treat them as two independent sources (like a "random walk" with 2 steps in statistics) you can use an RMS summation of the amplitudes.

Since the crossover at 4 kHz has both amplitudes as 0.7, or -3dB, the result is simply a flat line at 1 independent of frequency.

But it's a dangerous assumption and probably wrong for certain signals, so it's worth considering a worst case scenario, where the aliasing is perfectly correlated and in phase with the signal : in this case you can add amplitudes, giving a peak of 1.4 (+3dB) at 4 kHz.

With regard to: 


  Can you fix this problem by increasing the filter order? Not really. Figure 7 on page 10 shows the result of using a 6th-order Butterworth low pass filter instead of a 1st-order filter. We’ve improved the situation


Well he's right : you can't fix the problem simply by increasing filter order. You can improve it (decreasing the "alias" energy below 4 kHz and narrowing the +3dB peak in the worst case, thus decreasing its area). Thus you have improved the situation somewhat, but not fixed it. 

A fix : if you need to reduce the aliasing energy below some level : would involve reducing the filter cutoff frequency - as well as optionally increasing the order - such that the attenuation at Fs/2 (4 kHz here) met your target.

And regarding comments to the other answer : aliasing below 4 kHz is the system response (due to the sampling process) to energy after the filter above 4 kHz. (which btw sounds nasty). 

Aliasing at 7 kHz is huge because the filter has no attenuation at 1 kHz. It would also sound nasty but will be removed by the DAC's reconstruction filter.