Heres the tricky part. Instead of naturally grabbing the cork with the first finger of your right hand on the top of the cork and the thumb on the bottom, you turn your hands. Thus, the thumb of the right hand reaches under the first finger of the left hand to grab the top of the cork and the first finger reaches under the left thumb to grab the bottom of the cork.
Notice in the picture how the right thumb is reaching between the left first finger and thumb and the right first finger is reaching underneath the left thumb. You may have to turn both hands in the opposite directions to grab the second cork. The corks look as if they are interlocked, but theyre not.