It's a pretty simple idea:

if a 5g bullet moving at 330 m/s undergoes an elastic collision (i.e. no loss of kinetic energy) with an 80kg person initially at rest, then after the collision the person will be travelling at about 0.04 m/s (i.e. 4 cm/s) and the bullet will riccochet off at about 329.96 m/s. The target absorbs a mere 0.064 Joules of energy, the same amount as dropping a marble onto your chest from a height of a metre. As Anonymous coward above said (although he/she got the physics wrong too), simple physics - conservation of momentum and conservation of kinetic energy.

As is clear from the article, this won't work unless you fix the nanotube fibres at the ends, so your armour will have to be very stiff - think a mesh of nanotubes like flyscreen, attached at the points that they cross over, so that any nanotubes being hit by the bullet will pull sideways on the perpendicular nanotubes. The armour would be a shell, not a flexible fabric.

But these are trifling details of implementation, once the theory proves it can be done....