The general concept is that the individual capacitors in a series string of ultracapacitors do not all have precisely the same value of capacitance. However, they all generally have very close to the same maximum voltage limit.
When you put a given amount of charge Q on a capacitor, the voltage V across it is inversely proportional to its capacitance C, because of the relationship V = Q/C. In other words, a capacitor with a larger value will have less voltage on it.
In a series-connected string of capacitors, they all get the same charge, since they are all subjected to the same current for the same amount of time: Q = I × t.
What this means is that when charging up a string of ultracapacitors, you need to stop when the smallest-valued capacitor reaches its voltage limit, which means that all of the other larger-valued capacitors will have some unused capacity, since they have not yet reached their voltage limits.
The voltage equalization circuit allows you to make use of this unused capacity by shifting charge (by means of current pumps of one sort or another) from the lower-valued capacitors to the larger-valued capacitors during charging, and the other way during discharge. By making sure the individual capacitors are protected from overvoltage and undervoltage in this way, you can then treat the entire string as a single capacitor that has N× the voltage rating and 1/N the capacitance of the individual capacitors.