What are we seeing in this AC salt-water copper-wire experiment?
Is there really "a reaction of the copper with the salt in water" as the author of the video claims? Or is it a pure physical process where the copper "boils off" the wire?


Yes it reacts, no it doesn't boil. Essentially the wire corrodes at an accelerated rate where during the part of the cycle where the wire is the positive terminal the copper reacts as follows: 

$$\ce{Cu^0 + 2OH- -&gt; Cu^2+ + 2e- + 2OH- -&gt; Cu(OH)2 + 2e- -&gt; CuO + 2e- + H2O }$$
(expanded for clarity, not a reaction mechanism)

There is obviously oxygen generated too but I am only focusing on the copper part of the reaction. During the part of the cycle where the wire is the positive terminal the reverse of this reaction does not occur on the other half of the cycle and over time more and more copper will be consumed eventually consuming the entire wire.