Low weight split core transformer on 11kVA transmission line
That video is showing a current transformer or CT. With a fixed winding ratio (the primary side is just one turn) the output will be a current proportional to the current flowing in the primary conductor. No current in primary - no current in secondary.


  However, I fail to see how this is possible when a transformer cores which are small enough in size and weight go up to around 1kVa.


They are not trying to get 1 kVA to recharge a battery. They are only taking a few watts. That's what determines the transformer and electronics size. You could expect it to be of similar size to a wall-wart device adequate to charge the device in question.


  If I am correct an 11 kVA overhead transmissions line carries a current of around 100A. This gives a power rating needed for the transformer to be at least 1100kVA. This well exceeds the limit.


I think you are confusing units. You are talking kVA when you mean kV. The VA rating is just the line voltage by the line amps. If you know any two then you can calculate the third using \$ P = VI \$.


  If I am correct an 11 kVA overhead transmissions line carries a current of around 100 A. 


It could be any current between 0 (no load) to whatever the conductors, transformers and switchgear are rated for.


  This gives a power rating needed for the transformer to be at least 1100kVA. 


No it doesn't. The VA rating of any transformer is given by rated load voltage by load current.


  This well exceeds the limit.


The limit of what?



Most electronics want a constant voltage power supply. Using a CT will not give this and some tricky circuitry would be required to handle the varying current and try and get some useful power out of it.

Attaching such a device to a utility line would be theft and illegal.