What is the biological principle of this "holotropic breathwork" technique?
I think this is mostly caused by hyperventilation. The excessive breathing disturbs the balance between CO$_{2}$ and oxygen in our lungs. This will cause respiratory alkalosis (the blood pH, which is normally strictly regulated, gets higher), which can cause dizziness, headaches and fainting. 

The shift in pH can also disturb the eclectrolyte balance, especially the concentration of the free dissolved calcium, as it binds to negatively charged plasma proteins (which release protons into the blood due to the higher pH). This can lead to craps and spasms.
I would be very careful not playing around here, as this can have some severe consequences.