I think this is simply due to the fact that by cutting back you stimulate the renewal of vegetation. It may happen by bud-breaking of dormant buds. In a few words, you help the plant to get rid of the old vegetative organs. This technique is commonly adopted in many other trees, including those of horticultural interest.
Wounding of a willow tree such as pruning breaks whatever apical dominance is in effect and redistributes plant hormones such as auxins and cytokinins so that branching is promoted. Moreover, there may possibly be a response to ethylene exuded from the wound site, and this generally does promote vigorous growth as a defense mechanism to damage.
coppicing modifies aerial parts but cavities and root damage can be a long term side effect. at end on mass equation it is hard to evaluate a positive effect on carbon storage due to light competition and growing diseases. as far physiology there is a removal of apical dominance and a massive effect of ascending sap cytokinesis in betulas with rfererences to Bedenau and Auclair root damage study, in http://treephys.oxfordjournals.org/content/14/10/1149.full.pdf
Any external impact such as cutting back, triggers survival mechanisms who are expressed through the release of various chemicals within the plant, such as phytohormones, in order to compensate for losses.
New intensive growth is needed, so as for the plant to be in a position to provide nutrients for that growth and the already existing one, so the physiology is sped up.
Besides above ground, there is stimulation to the root system as well, for uptaking nutrients necessary for such task. For trees that don't bare fruit, xylem growth is more intense, than fruit baring. The latter can reproduced by seeds in the fruit, so has to be a balance between xylem multiplication and seed(fruit), which will provide the survival of the species. Beauty of nature.
To add to the answers already given, there will surely be a tendency (implemented by appropriate production and translocation of plant growth substances) to restore the balance between different parts of a tree. Given the existing root system and whatever remains of the lower stem, accelerated stem growth will be the response that tends to restore the balance. And, as already indicated, such stimulated growth is typically entirely vegetative at the start of the response, without the diversion into reproduction and shedding of reproductive structures. Also, the reduced height is likely to reduce certain energy costs of translocation to greater heights with associated impacts on water potentials, etc. (although I am not conversant with the quantitative aspects of this).