What this guy is doing is letting the harmonic ring, placing his right hand on the upper side of the guitar body, and then placing his left hand underneath the neck without touching the strings (so as the harmonic does not stop ringing). He then pushes downwards with his right hand, pulls up with his left hand.
This bends the neck slightly inwards, lowering the tension on the strings, creating the pitch alteration that you hear. This is a similar principle to how the whammy bar works, except instead of bending the neck to lower string tension, the bridge is moved.
Bending a guitar neck is anyway is not something I would recommend, as it can cause damage to the neck-body join, amongst other things. As long as it is not done too harshly and too regularly, it shouldn't cause massive permanent damage.
Hope this helps.
If you are interested, check out this question for how a type of electric guitar tremolo works.