Perhaps like said in the video it had different voltage potential, so it was like static electricity, when he brings the rod to the wire the electrons equalize between the wire and the helicopter and don't move anymore therefore not creating sparks?
That's basically it, although the system is not quite static. If the wires are carrying (say) 250 kV AC, then the potential of the wire is changing from +354 kV to -354 kV, 60 times a second. (354 kV is the peak voltage of a 250 kV RMS sine wave.) When the helicopter is touching the wire, then in order to equalize the potentials the helicopter has to change from +354 kV to -354 kV in step with the wire, and to achieve this current has to flow back and forth between the wire and the helicopter. The helicopter is acting as a capacitor, or storage container for charge, and current has to flow to make a capacitor change its voltage.
Since the potential on the wire is constantly changing with the AC voltage the sparks would fly forever and never stop unless they bonded the helicopter to the wire with the metal clamp in the way they showed.
(You may ask, how does the wire "know" the helicopter is at a different potential when it is isolated in mid-air like that? The answer is that the high voltage wire has previously had it's voltage referenced to ground at the transformer station by an earth bond on the neutral conductor. Tracing back in time the helicopter also was touching the ground before it took off, so the helicopter started out at ground potential with reference to the high voltage wire. Since the helicopter is acting like a capacitor, it has remembered it's potential in the same way that capacitors hold on to voltage.)